IMPORTANT NOTE: The questions must be addressed in its full context. These questions are an opportunity to go outside the box to demonstrate your analytical, integrative, problem- solving and critical thinking skills using the knowledge acquired in your readings. As a result, it is very important to pay close attention to the questions and be able to conduct your discussions in the context of your question.  – Please keep this in mind when you complete this assignment.You must expand your ideas further. Analysis must be deep and very instructive. ANSWER THE FOLLOWING QUESTIONS. Each question should be answered in at least 300 words. Quality of content and use of course and outside-of-course resources to support your position or analysis. The answers should not be in the form of essay, just straight to the point- Work must be original and cite your sources.Please be sure to answer the question completely but specifically in well-written complete sentences. Use the attached lecture to help you answer these questions, and conduct your own research1. From the Chapter 7 discussion question you have settled upon one of the two proposed projects.  Now you need to understand the requirements associated with capital budgeting for this project as you move forward.  Based upon your study of material in Chapter 8, what are the major steps involved in the capital budgeting process?2. In your study of the capital budgeting process it is obvious that you need to understand the concept of free cash flow related to the project selected.  Discuss the adjustments that will need to be made to the estimated free cash flow in your evaluation of the selected project.  Include in your discussion comments as to the roll that salvage value of equipment would play in your estimate of free cash flow.financial_decision_making_and_the_law_of_one_price.docxFinancial Decision Making and the
Law of One Price
I
N MID-2007, MICROSOFT DECIDED TO ENTER A BIDDING WAR with competitors Google and Yahoo! for a stake
in the fast-growing social networking site, Facebook. How did Microsoft’s managers decide that this was a good decision?
Every decision has future consequences that will affect the value of the firm. These consequences will generally include both
benefits and costs. For example, after raising its offer, Microsoft ultimately succeeded in buying a 1.6% stake in Facebook, along
with the right to place banner ads on the Facebook Web site, for $240 million. In addition to the upfront cost of $240 million,
Microsoft also incurred ongoing costs associated with software development for the platform, network infrastructure, and
international marketing efforts to attract advertisers. The benefits of the deal to Microsoft included the revenues associated with the
advertising sales, together with the appreciation of its 1.6% stake in Facebook. In the end, Microsoft’s decision appeared to be a
good one—in addition to advertising benefits, by the time of Facebook’s IPO in May 2012, the value of Microsoft’s 1.6% stake had
grown to over $1 billion.
More generally, a decision is good for the firm’s investors if it increases the firm’s value by providing benefits whose value
exceeds the costs. But comparing costs and benefits is often complicated because they occur at different points in time, may be in
different currencies, or may have differ- ent risks associated with them.To make a valid comparison, we must use the tools of
finance to express all costs and benefits in common terms. In this chapter, we introduce a central principle of finance, which we
name the Valuation Principle, which states that we can use current market prices to determine the value today of the costs and
benefits associated with a decision.This principle allows us to apply the concept of net present value (NPV) as a way to compare
the costs and benefits of a project in terms of a common unit—namely, dollars today. We will then be able to evaluate a decision by
answering this question: Does the cash value today of its
CHAPTER
3
NOTATION
59
NPV net present value
rf risk-free interest rate
PV present value
59
60 Chapter 3 Financial Decision Making and the Law of One Price
benefits exceed the cash value today of its costs? In addition, we will see that the NPV indicates the net amount by which the
decision will increase wealth.
We then turn to financial markets and apply these same tools to determine the prices of securities that trade in the market. We
discuss strategies called arbitrage, which allow us to exploit situations in which the prices of publicly available invest- ment
opportunities do not conform to these values. Because investors trade rapidly to take advantage of arbitrage opportunities, we argue
that equivalent investment opportunities trading simultaneously in competitive markets must have the same price.This Law of One
Price is the unifying theme of valuation that we use throughout this text.
3.1 Valuing Decisions
A financial manager’s job is to make decisions on behalf of the firm’s investors. For example, when
faced with an increase in demand for the firm’s products, a manager may need to decide whether to raise
prices or increase production. If the decision is to raise production and a new facility is required, is it
better to rent or purchase the facility? If the facility will be purchased, should the firm pay cash or
borrow the funds needed to pay for it?
In this book, our objective is to explain how to make decisions that increase the value of the firm to its
investors. In principle, the idea is simple and intuitive: For good decisions, the benefits exceed the costs.
Of course, real-world opportunities are usually complex and so the costs and benefits are often difficult
to quantify. The analysis will often involve skills from other management disciplines, as in these
examples:
Marketing : to forecast the increase in revenues resulting from an advertising campaign
Accounting : to estimate the tax savings from a restructuring
Economics: to determine the increase in demand from lowering the price of a product
Organizational Behavior: to estimate the productivity gains from a change in manage- ment structure
Strategy: to predict a competitor’s response to a price increase Operations: to estimate the cost savings
from a plant modernization
For the remainder of this text, we assume that the analysis of these other disciplines has been completed
to quantify the costs and benefits associated with a decision. With that task done, the financial manager
must compare the costs and benefits and determine the best decision to make for the value of the firm.
Analyzing Costs and Benefits
The first step in decision making is to identify the costs and benefits of a decision. The next step is to
quantify these costs and benefits. In order to compare the costs and benefits, we need to evaluate them in
the same terms—cash today. Let’s make this concrete with a simple example.
Suppose a jewelry manufacturer has the opportunity to trade 400 ounces of silver for 10 ounces of gold
today. Because an ounce of gold differs in value from an ounce of silver,
it is incorrect to compare 400 ounces to 10 ounces and conclude that the larger quantity is better. Instead,
to compare the costs and benefits, we first need to quantify their values in equivalent terms.
Consider the silver. What is its cash value today? Suppose silver can be bought and sold for a current
market price of $15 per ounce. Then the 400 ounces of silver we give up has a cash value of1
(400 ounces of silver today) * ($15/ounce of silver today) = $6000 today If the current market price for
gold is $900 per ounce, then the 10 ounces of gold we
receive has a cash value of
(10 ounces of gold today) * ($900/ounce of gold today) = $9000 today
Now that we have quantified the costs and benefits in terms of a common measure of value, cash today,
we can compare them. The jeweler’s opportunity has a benefit of $9000 today and a cost of $6000 today,
so the net value of the decision is $9000 – $6000 = $3000 today. By accepting the trade, the jewelry firm
will be richer by $3000.
Using Market Prices to Determine Cash Values
In evaluating the jeweler’s decision, we used the current market price to convert from ounces of silver or
gold to dollars. We did not concern ourselves with whether the jeweler thought that the price was fair or
whether the jeweler would use the silver or gold. Do such considerations matter? Suppose, for example,
that the jeweler does not need the gold, or thinks the current price of gold is too high. Would he value the
gold at less than $9000? The answer is no—he can always sell the gold at the current market price and
receive $9000 right now. Similarly, he would not value the gold at more than $9000, because even if he
really needs the gold or thinks the current price of gold is too low, he can always buy 10 ounces of gold
for $9000. Thus, independent of his own views or preferences, the value of the gold to the jeweler is
$9000.
This example illustrates an important general principle: Whenever a good trades in a competitive
market—by which we mean a market in which it can be bought and sold at the same price—that price
determines the cash value of the good. As long as a competi- tive market exists, the value of the good
will not depend on the views or preferences of the decision maker.
3.1 Valuing Decisions 61
Competitive Market Prices Determine Value
Problem
You have just won a radio contest and are disappointed to find out that the prize is four tickets to the Def Leppard reunion tour (face val
each). Not being a fan of 1980s power rock, you have no intention of going to the show. However, there is a second choice: two tickets
favorite band’s sold-out show (face value $45 each). You notice that on eBay, tickets to the Def Leppard show are being bought and sol
apiece and tickets to your favorite band’s show are being bought and sold at $50 each. Which prize should you choose?
EXAMPLE 3.1
1
You might worry about commissions or other transactions costs that are incurred when buying or selling gold, in addition to the
market price. For now, we will ignore transactions costs, and discuss their effect in the appendix to this chapter.
62
Chapter 3 Financial Decision Making and the Law of One Price
Solution
Competitive market prices, not your personal preferences (nor the face value of the tickets), are relevant here:
Four Def Leppard tickets at $30 apiece = $120 market value
market value
Two of your favorite band’s tickets at $50 apiece = $100
Instead of taking the tickets to your favorite band, you should accept the Def Leppard tickets, sell them on eBay, and use
the proceeds to buy two tickets to your favorite band’s show. You’ll even have $20 left over to buy a T-shirt.
Thus, by evaluating cost and benefits using competitive market prices, we can deter- mine whether a
decision will make the firm and its investors wealthier. This point is one of the central and most powerful
ideas in finance, which we call the Valuation Principle:
The value of an asset to the firm or its investors is determined by its competitive market price. The
benefits and costs of a decision should be evaluated using these market prices, and when the value of the
benefits exceeds the value of the costs, the decision will increase the market value of the firm.
The Valuation Principle provides the basis for decision making throughout this text. In the remainder of
this chapter, we first apply it to decisions whose costs and benefits occur at different points in time and
develop the main tool of project evaluation, the Net Present Value Rule. We then consider its
consequences for the prices of assets in the market and develop the concept of the Law of One Price.
Applying the Valuation Principle
Problem
You are the operations manager at your firm. Due to a pre-existing contract, you have the oppor- tunity to acquire 200 barrels
3000 pounds of copper for a total of $12,000. The cur- rent competitive market price of oil is $50 per barrel and for copper is
pound. You are not sure you need all of the oil and copper, and are concerned that the value of both commodities may fall in
Should you take this opportunity?
Solution
To answer this question, you need to convert the costs and benefits to their cash values using market prices:
(200 barrels of oil) * ($50/barrel of oil today) = $10,000 today
(3000 pounds of copper) * ($2/pound of copper today) = $6000 today
The net value of the opportunity is $10,000 + $6000 – $12,000 = $4000 today. Because the net value is positive, you should ta
value depends only on the current market prices for oil and copper. Even if you do not need all the oil or copper, or expect th
to fall, you can sell them at current market prices and obtain their value of $16,000. Thus, the opportunity is a good one for th
will increase its value by $4000.
EXAMPLE 3.2
3.2 Interest Rates and the Time Value of Money 63
When Competitive Market Prices Are Not Available
Competitive market prices allow us to calculate the value of a decision without worrying about the tastes or opinions of the
maker. When competitive prices are not available, we can no longer do this. Prices at retail stores, for example, are one sid
can buy at the posted price, but you cannot sell the good to the store at that same price. We cannot use these one-sided pric
determine an exact cash value. They determine the maximum value of the good (since it can always be purchased at that pr
indi- vidual may value it for much less depending on his or her preferences for the good.
Let’s consider an example. It has long been common for banks to entice new depositors by offering free gifts for opening a
account. In 2012, ThinkForex offered a free
iPad 3 for individuals opening a new account. At the time, the retail price of that model iPad was $539. But because there i
competitive market to trade iPads, the value of the iPad depends on whether you were going to buy one or not.
If you planned to buy the iPad anyway, then the value to you is $539, the price you would otherwise pay for it. But if you d
want or need the iPad, the value of the offer would depend on the price you could get for the iPad. For example, if you cou
iPad for $450 to your friend, then ThinkForex’s offer is worth $450 to you. Thus, depending on your preferences, ThinkFo
is worth somewhere between $450 (you don’t want an iPad) and $539 (you definitely want one).
CONCEPT CHECK
3.2
1. In order to compare the costs and benefits of a decision, what must we determine? 2. If crude oil trades in a competitive market,
would an oil refiner that has a use for the
oil value it differently than another investor?
Interest Rates and the Time Value of Money
For most financial decisions, unlike in the examples presented so far, costs and benefits occur at different
points in time. For example, typical investment projects incur costs upfront and provide benefits in the
future. In this section, we show how to account for this time difference when evaluating a project.
The Time Value of Money
Consider an investment opportunity with the following certain cash flows:
Cost: $100,000 today Benefit: $105,000 in one year
Because both are expressed in dollar terms, it might appear that the cost and benefit are directly
comparable so that the project’s net value is $105,000 – $100,000 = $5000. But this calculation ignores
the timing of the costs and benefits, and it treats money today as equivalent to money in one year.
In general, a dollar today is worth more than a dollar in one year. If you have $1 today, you can invest it.
For example, if you deposit it in a bank account paying 7% interest, you will have $1.07 at the end of one
year. We call the difference in value between money today and money in the future the time value of
money.
The Interest Rate: An Exchange Rate Across Time
By depositing money into a savings account, we can convert money today into money in the future with
no risk. Similarly, by borrowing money from the bank, we can exchange money in the future for money
today. The rate at which we can exchange money today for money in the future is determined by the
current interest rate. In the same way that
64 Chapter 3 Financial Decision Making and the Law of One Price
an exchange rate allows us to convert money from one currency to another, the interest rate allows us to
convert money from one point in time to another. In essence, an inter- est rate is like an exchange rate
across time. It tells us the market price today of money in the future.
Suppose the current annual interest rate is 7%. By investing or borrowing at this rate, we can exchange
$1.07 in one year for each $1 today. More generally, we define the risk-free interest rate, rf , for a given
period as the interest rate at which money can be borrowed or lent without risk over that period. We can
exchange (1 + rf ) dollars in the future per dollar today, and vice versa, without risk. We refer to (1 + rf )
as the interest rate factor for risk- free cash flows; it defines the exchange rate across time, and has
units of “$ in one year/$ today.”
As with other market prices, the risk-free interest rate depends on supply and demand. In particular, at
the risk-free interest rate the supply of savings equals the demand for bor- rowing. After we know the
risk-free interest rate, we can use it to evaluate other decisions in which costs and benefits are separated
in time without knowing the investor’s preferences.
Value of Investment in One Year. Let’s reevaluate the investment we considered ear- lier, this time taking
into account the time value of money. If the interest rate is 7%, then we can express our costs as
Cost = ($100,000 today) * (1.07 $ in one year/$ today) = $107,000 in one year
Think of this amount as the opportunity cost of spending $100,000 today: We give up the $107,000 we
would have had in one year if we had left the money in the bank. Alterna- tively, if we were to borrow
the $100,000, we would owe $107,000 in one year.
Both costs and benefits are now in terms of “dollars in one year,” so we can compare them and compute
the investment’s net value:
$105,000 – $107,000 = -$2000 in one year
In other words, we could earn $2000 more in one year by putting our $100,000 in the bank rather than
making this investment. We should reject the investment: If we took it, we would be $2000 poorer in one
year than if we didn’t.
Value of Investment Today. The previous calculation expressed the value of the costs and benefits in terms
of dollars in one year. Alternatively, we can use the interest rate fac- tor to convert to dollars today.
Consider the benefit of $105,000 in one year. What is the equivalent amount in terms of dollars today?
That is, how much would we need to have in the bank today so that we would end up with $105,000 in
the bank in one year? We find this amount by dividing by the interest rate factor:
1
Benefit = ($105,000 in one year) , (1.07 $ in one year/$ today) = $105,000 * today
This is also the amount the bank would lend to us today if we promised to repay $105,000 in one year.2
Thus, it is the competitive market price at which we can “buy” or “sell” $105,000 in one year.
2
We are assuming the bank will both borrow and lend at the risk-free interest rate. We discuss the case when these rates differ in
“Arbitrage with Transactions Costs” in the appendix to this chapter.
1.07 = $98,130.84 today
3.2 Interest Rates and the Time Value of Money 65
Now we are ready to compute the net value of the investment: $98,130.84 – $100,000 = -$1869.16 today
Once again, the negative result indicates that we should reject the investment. Taking the investment
would make us $1869.16 poorer today because we have given up $100,000 for something worth only
$98,130.84.
Present Versus Future Value. This calculation demonstrates that our decision is the same whether we
express the value of the investment in terms of dollars in one year or dol- lars today: We should reject the
investment. Indeed, if we convert from dollars today to dollars in one year,
(-$1869.16 today) * (1.07 $ in one year/$ today) = -$2000 in one year
we see that the two results are equivalent, but expressed as values at different points in time. When we
express the value in terms of dollars today, we call it the present value (PV) of the investment. If we
express it in terms of dollars in the future, we call it the future value (FV) of the investment.
Discount Factors and Rates. When computing a present value as in the preceding calculation, we can
interpret the term
1
1
= = 0.93458 $ today/$ in one year 1+r 1.07
as the price today of $1 in one year. Note that the value is less than $1—money in the
future is worth less today, and so its price reflects a discount. Because it provides the discount at which we can purchase money in the future, the amount 1 is called the one- 1+r
year discount factor. The risk-free interest rate is also referred to as the discount rate for a risk-free
investment.
Comparing Costs at Different Points in Time
Problem
The cost of rebuilding the San Francisco Bay Bridge to make it earthquake-safe was approxi- mately $3 billion in 20
time, engineers estimated that if the project were delayed to 2005, the cost would rise by 10%. If the interest rate we
what would be the cost of a delay in terms of dollars in 2004?
Solution
If the project were delayed, it would cost $3 billion * (1.10) = $3.3 billion in 2005. To compare this amount to the co
billion in 2004, we must convert it using the interest rate of 2%:
$3.3 billion in 2005 , ($1.02 in 2005/$ in 2004) = $3.235 billion in 2004 Therefore, the cost of a delay of one year w
$3.235 billion – $3 billion = $235 million in 2004
million in cash.
That is, delaying the project for one year was equivalent to givin
EXAMPLE 3.3
We can use the risk-free interest rate to determine values in the same way we used competitive market
prices. Figure 3.1 illustrates how we use competitive market prices, exchange rates, and interest rates to
convert between dollars today and other goods, curren- cies, or dollars in the future.
66 Chapter 3 Financial Decision Making and the Law of One Price
FIGURE 3.1
Gold Price ($/oz)
Gold Price ($/oz)
Exchange Rate (€/$)
Exchange Rate (€/$)
(1 rf )
(1 rf )
Euros Today
Dollars in One Year
Dollars Today
Ounces of Gold Today
Converting between Dollars Today and Gold, Euros, or Dollars in
We can convert dollars today to different goods, currencies, or points
rate, or interest rate.
the Future
in time by using the competitive market price, exchange
CONCEPT CHECK
3.3
1. How do you compare costs at different points in time?
money in one
2. If interest rates rise, what happens to the value today of a promise of
year?
Present Value and the NPV Decision Rule
In Section 3.2, we converted between cash today and cash in the future using the risk-free interest rate.
As long as we convert costs and benefits to the same point in time, we can compare them to make a
decision. In practice, however, most corporations prefer to mea- sure values in terms of their present
value—that is, in terms of cash today. In this section we apply the Valuation Principle to derive the
concept of the net present value, or NPV, and define the “golden rule” of financial decision making, the
NPV Rule.
Net Present Value
When we compute the value of a cost or benefit in terms of cash today, we refer to it as the present value
(PV). Similarly, we define the net present value (NPV) of a project or investment as the difference
between the present value of its benefits and the present value of its costs:
Net Present Value
NPV = PV(Benefits) – PV(Costs) (3.1)
If we use positive cash flows to represent benefits and negative cash flows to represent costs, and
calculate the present value of multiple cash flows as the sum of present values for individual cash flows,
we can also write this definition as
NPV = PV(All project cash flows) (3.2) That is, the NPV is the total of the present values of all project
cash flows.
3.3 Present Value and the NPV Decision Rule 67
Let’s consider a simple example. Suppose your firm is offered the following investment opportunity: In
exchange for $500 today, you will receive $550 in one year with certainty. If the risk-free interest rate is
8% per year then
PV (Benefit) = ($550 in one year) , (1.08 $ in one year/$ today) = $509.26 today
This PV is the amount we would need to put in the bank today to generate $550 in one year ($509.26 *
1.08 = $550). In other words, the present value is the cash cost today of “doing it yourself ”—it is the
amount you need to invest at the current interest rate to recreate the cash flow.
Once the costs and benefits are in present value terms, we can compute the investment’s NPV:
NPV = $509.26 – $500 = $9.26 today
But what if your firm doesn’t have the $500 needed to cover the initial cost of the project? Does the
project still have the same value? Because we computed the value using competitive market prices, it
should not depend on your tastes or the amount of cash your firm has in the bank. If your firm doesn’t
have the $500, it could borrow $509.26 from the bank at the 8% interest rate and then take the project.
What are your cash flows in this case?
Today: $509.26 (loan) – $500 (invested in the project) = $9.26 In one year: $550 (from project) – $509.26
* 1.08 (loan balance) = $0
This transaction leaves you with exactly $9.26 extra cash today and no future net obliga- tions. So taking
the project is like having an extra $9.26 in cash up front. Thus, the NPV expresses the value of an
investment decision as an amount of cash received today. As long as the NPV is positive, the decision
increases the value of the firm and is a good decision regard- less of your current cash needs or
preferences regarding when to spend the money.
The NPV Decision Rule
Because NPV is expressed in terms of cash today, it simplifies decision making. As long as we have
correctly captured all of the costs and benefits of the project, decisions with a positive NPV will increase
the wealth of the firm and its investors. We capture this logic in the NPV Decision Rule:
When making an investment decision, take the alternative with the highest NPV. Choosing this
alternative is equivalent to receiving its NPV in cash today.
Accepting or Rejecting a Project. A common financial decision is whether to accept or reject a project.
Because rejecting the project generally has NPV = 0 (there are no new costs or benefits from not doing
the project), the NPV decision rule implies that we should
Accept those projects with positive NPV because accepting them is equivalent to receiv- ing their NPV
in cash today, and
I
Reject those projects with negative NPV; accepting them would reduce the wealth of investors, whereas
not doing them has no cost (NPV = 0).
I
If the NPV is exactly zero, you will neither gain nor lose by accepting the project rather than rejecting it.
It is not a bad project because it does not reduce firm value, but it does not increase value either.
68
Chapter 3 Financial Decision Making and the Law of One Price EXAMPLE 3.4
The NPV Is Equivalent to Cash Today
Problem
Your firm needs to buy a new $9500 copier. As part of a promotion, the manufacturer has offered to let you pay $10,000 i
rather than pay cash today. Suppose the risk-free interest rate is 7% per year. Is this offer a good deal? Show that its NPV
cash in your pocket.
Solution
If you take the offer, the benefit is that you won’t have to pay $9500 today, which is already in PV terms. The cost, howev
$10,000 in one year. We therefore convert the cost to a present value at the risk-free interest rate:
PV(Cost) = ($10,000 in one year) , (1.07 $ in one year/$ today) = $9345.79 today The NPV of the promotional offer is the
between the benefits and the costs:
NPV = $9500 – $9345.79 = $154.21 today
The NPV is positive, so the investment is a good deal. It is equivalent to getting a cash discount today of $154.21, and onl
$9345.79 today for the copier. To confirm our calculation, suppose you take the offer and invest $9345.79 in a bank payin
interest. With interest, this amount will grow to $9345.79 * 1.07 = $10,000 in one year, which you can use to pay for the c
Choosing among Alternatives. We can also use the NPV decision rule to choose among projects. To do so,
we must compute the NPV of each alternative, and then select the one with the highest NPV. This
alternative is the one that will lead to the largest increase in the value of the firm.
Choosing among Alternative Plans
Problem
Suppose you started a Web site hosting business and then decided to return to school. Now that you are back in school, yo
considering selling the business within the next year. An investor has offered to buy the business for $200,000 whenever y
ready. If the interest rate is 10%, which of the following three alternatives is the best choice?
1. Sell the business now.
2. Scale back the business and continue running it while you are in school for one more year, and then sell the business (re
you to spend $30,000 on expenses now, but generat- ing $50,000 in profit at the end of the year).
3. Hire someone to manage the business while you are in school for one more year, and then sell the business (requiring yo
$50,000 on expenses now, but generating $100,000 in profit at the end of the year).
Solution
The cash flows and NPVs for each alternative are calculated in Table 3.1. Faced with these three alternatives, the best cho
one with highest NPV: Hire a manager and sell in one year. Choosing this alternative is equivalent to receiving $222,727 t
EXAMPLE 3.5
3.3 Present Value and the NPV Decision Rule 69
TABLE 3.1
Sell Now
Scale Back Operations
Hire a Manager
Today
$200,000 -$30,000
-$50,000
In One Year
0 $50,000 $200,000
$100,000 $200,000
NPV
$200,000
-$30,000 +
$250,000
= $197,273
1.10
-$50,000 +
$300,000
= $222,727 1.10
Cash Flows and NPVs for Web Site Business Alternatives
NPV and Cash Needs
When we compare projects with different patterns of present and future cash flows, we may have
preferences regarding when to receive the cash. Some may need cash today; others may prefer to save for
the future. In the Web site hosting business example, hiring a manager and selling in one year has the
highest NPV. However, this option requires an initial outlay of $50,000, as opposed to selling the
business and receiving $200,000 immediately. Suppose you need $60,000 in cash now to pay for school
and other expenses. Would selling the business be a better choice in that case?
As was true for the jeweler considering trading silver for gold in Section 3.1, the answer is again no. As
long as you can borrow and lend at the 10% interest rate, hiring a manager is the best choice whatever
your preferences regarding the timing of the cash flows. To see why, suppose you borrow $110,000 at
the rate of 10% and hire the manager. Then you will owe $110,000 * 1.10 = $121,000 in one year, for
total cash flows shown in Table 3.2. Compare these cash flows with those from selling now, and
investing the excess $140,000 (which, at the rate of 10%, will grow to $140,000 * 1.10 = $154,000 in one
year). Both strategies provide $60,000 in cash today, but the combination of hiring a manager and
borrowing generates an additional $179,000 – $154,000 = $25,000 in one year.3 Thus, even if you need
$60,000 now, hiring the manager and selling in one year is still the best option.
TABLE 3.2
Cash Flows of Hiring and Borrowing Versus Selling and Investing
Today
Hire a Manager -$50,000 Borrow $110,000
In One Year
$300,000 – $121,000 $179,000
$0 $154,000
$154,000
Total Cash Flow Versus
Sell Now
Invest
Total Cash Flow
$60,000 $200,000
– $140,000 $60,000
3
Note also that the present value of this additional cash flow, $25,000 , 1.10 = $22,727, is exactly the difference in NPVs between
the two alternatives.
70 Chapter 3 Financial Decision Making and the Law of One Price This example illustrates the following general
principle:
CONCEPT CHECK
3.4
Regardless of our preferences for cash today versus cash in the future, we should always maximize NPV
first. We can then borrow or lend to shift cash flows through time and find our most preferred pattern of
cash flows.
1. What is the NPV decision rule?
2. Why doesn’t the NPV decision rule depend on the investor’s preferences?
Arbitrage and the Law of One Price
So far, we have emphasized the importance of using competitive market prices to compute the NPV. But
is there always only one such price? What if the same good trades for dif- ferent prices in different
markets? Consider gold. Gold trades in many different markets, with the largest markets in New York
and London. To value an ounce of gold we could look up the competitive price in either of these markets.
But suppose gold is trading for $850 per ounce in New York and $900 per ounce in London. Which price
should we use?
Fortunately, such situations do not arise, and it is easy to see why. Recall that these are competitive
market prices at which you can both buy and sell. Thus, you can make money in this situation simply by
buying gold for $850 per ounce in New York and then imme- diately selling it for $900 per ounce in
London.4 You will make $900 – $850 = $50 per ounce for each ounce you buy and sell. Trading 1 million
ounces at these prices, you would make $50 million with no risk or investment! This is a case where that
old adage, “Buy low, sell high,” can be followed perfectly.
Of course, you will not be the only one making these trades. Everyone who sees these prices will want to
trade as many ounces as possible. Within seconds, the market in New York would be flooded with buy
orders, and the market in London would be flooded with sell orders. Although a few ounces (traded by
the lucky individuals who spotted this opportunity first) might be exchanged at these prices, the price of
gold in New York would quickly rise in response to all the orders, and the price in London would rapidly
fall.5 Prices would continue to change until they were equalized somewhere in the middle, such as $875
per ounce.
Arbitrage
The practice of buying and selling equivalent goods in different markets to take advantage of a price
difference is known as arbitrage. More generally, we refer to any situation in which it is possible to
make a profit without taking any risk or making any investment as an arbitrage opportunity. Because
an arbitrage opportunity has a positive NPV, whenever an arbitrage opportunity appears in financial
markets, investors will race to take advantage
4
There is no need to transport the gold from New York to London because investors in these markets trade ownership rights to gold
that is stored securely elsewhere. 5As economists would say, supply would not equal demand in these markets. In New York,
demand would be infinite because everyone would want to buy. For equilibrium to be restored so that supply equals
demand, the price in New York would have to rise. Similarly, in London there would be infinite supply until the price there fell.
3.4 Arbitrage and the Law of One Price 71
NASDAQ SOES Bandits
The NASDAQ stock market differs from other markets such as the NYSE in that it includes multiple dealers who all trade the same
stock. For example, on a given day, as many as ten or more dealers may post prices at which they are will- ing to trade Apple Comput
stock (AAPL). The NASDAQ also has a Small Order Execution System (SOES) that allows individual investors to execute trades
directly with a market maker instantly through an electronic system.
When SOES was first launched in the late 1980s, a type of trader referred to as a “SOES bandit” emerged. These traders watched the
quotes of different dealers, waiting for arbitrage opportunities to arise. For example, if one dealer was offering to sell AAPL at $580.2
and another was willing to buy at
$580.30, the SOES bandit could profit by instantly buying 1000 shares at $580.25 from the first dealer and selling 1000 shares at
$580.30 to the second dealer. Such a trade would yield an arbitrage profit of 1000 * $0.05 = $50.
In the past, by making trades like this one many times per day, these traders could make a reasonable amount of money. Before long,
activity of these traders forced deal- ers to monitor their own quotes much more actively so as to avoid being “picked off ” by these
bandits. Today, this sort of arbitrage opportunity rarely appears.*
*See J. Harris and P. Schultz, “The Trading Profits of SOES Bandits,” Journal of Financial Economics 50(2), (October 1998): 39–62.
CONCEPT CHECK
of it. Those investors who spot the opportunity first and who can trade quickly will have the ability to
exploit it. Once they place their trades, prices will respond, causing the arbi- trage opportunity to
evaporate.
Arbitrage opportunities are like money lying in the street; once spotted, they will quickly disappear. Thus
the normal state of affairs in markets should be that no arbitrage opportu- nities exist. We call a
competitive market in which there are no arbitrage opportunities a normal market.6
Law of One Price
In a normal market, the price of gold at any point in time will be the same in London and New York. The
same logic applies more generally whenever equivalent investment opportunities trade in two different
competitive markets. If the prices in the two markets differ, investors will profit immediately by buying
in the market where it is cheap and selling in the market where it is expensive. In doing so, they will
equalize the prices. As a result, prices will not differ (at least not for long). This important property is the
Law of One Price:
If equivalent investment opportunities trade simultaneously in different competitive mar- kets, then they
must trade for the same price in both markets.
One useful consequence of the Law of One Price is that when evaluating costs and ben- efits to compute
a net present value, we can use any competitive price to determine a cash value, without checking the
price in all possible markets.
1. If the Law of One Price were violated, how could investors profit?
their actions affect prices?
2. When investors exploit an arbitrage opportunity, how do
6
The term efficient market is also sometimes used to describe a market that, along with other properties, is without arbitrage
opportunities. We avoid the term because it is often vaguely (and inconsistently) defined.
72 Chapter 3 Financial Decision Making and the Law of One Price
3.5 No-Arbitrage and
Security Prices
An investment opportunity that trades in a financial market is known as a financial security (or, more
simply, a security). The notions of arbitrage and the Law of One Price have important implications for
security prices. We begin exploring its implications for the prices of individual securities as well as
market interest rates. We then broaden our perspective to value a package of securities. Along the way,
we will develop some important insights about firm decision making and firm value that will underpin
our study through- out this textbook.
Valuing a Security with the Law of One Price
The Law of One Price tells us that the prices of equivalent investment opportunities should be the same.
We can use this idea to value a security if we can find another equivalent investment whose price is
already known. Consider a simple security that promises a one- time payment to its owner of $1000 in
one year’s time. Suppose there is no risk that the payment will not be made. One example of this type of
security is a bond, a security sold by governments and corporations to raise money from investors today
in exchange for the promised future payment. If the risk-free interest rate is 5%, what can we conclude
about the price of this bond in a normal market?
To answer this question, consider an alternative investment that would generate the same cash flow as
this bond. Suppose we invest money at the bank at the risk-free interest rate. How much do we need to
invest today to receive $1000 in one year? As we saw in Section 3.3, the cost today of recreating a future
cash flow on our own is its present value:
PV ($1000 in one year) = ($1000 in one year) , (1.05 $ in one year/$ today) = $952.38 today
If we invest $952.38 today at the 5% risk-free interest rate, we will have $1000 in one year’s time with
no risk.
We now have two ways to receive the same cash flow: (1) buy the bond or (2) invest $952.38 at the 5%
risk-free interest rate. Because these transactions produce equivalent cash flows, the Law of One Price
implies that, in a normal market, they must have the same price (or cost). Therefore,
Price (Bond) = $952.38
An Old Joke
There is an old joke that many finance professors enjoy telling their students. It goes like this:
A finance professor and a student are walking down a street. The student notices a $100 bill lying on the pavement and leans down to pick it
up. The finance professor immediately intervenes and says, “Don’t bother; there is no free lunch. If that were a real $100 bill lying there,
somebody would already have picked it up!”
This joke invariably generates much laughter because it makes fun of the principle of no arbitrage in competitive markets. But once the
laughter dies down, the professor
then asks whether anyone has ever actually found a real $100 bill lying on the pavement. The ensuing silence is the real lesson behind the
joke.
This joke sums up the point of focusing on markets in which no arbitrage opportunities exist. Free $100 bills lying on the pavement, like
arbitrage opportunities, are extremely rare for two reasons: (1) Because $100 is a large amount of money, people are especially careful not to
lose it, and (2) in the rare event when someone does inadvertently drop $100, the likelihood of your finding it before someone else does is
extremely small.
3.5 No-Arbitrage and Security Prices 73
TABLE 3.3
Buy the bond
Borrow from the bank Net cash flow
Net Cash Flows from Buying the Bond and Borrowing
Today ($)
-940.00 +952.38 +12.38
In One Year ($)
+1000.00 -1000.00 0.00
Identifying Arbitrage Opportunities with Securities. Recall that the Law of One Price is based on the
possibility of arbitrage: If the bond had a different price, there would be an arbitrage opportunity. For
example, suppose the bond traded for a price of $940. How could we profit in this situation?
In this case, we can buy the bond for $940 and at the same time borrow $952.38 from the bank. Given
the 5% interest rate, we will owe the bank $952.38 * 1.05 = $1000 in one year. Our overall cash flows
from this pair of transactions are as shown in Table 3.3. Using this strategy we can earn $12.38 in cash
today for each bond that we buy, without taking any risk or paying any of our own money in the future.
Of course, as we—and others who see the opportunity—start buying the bond, its price will quickly rise
until it reaches $952.38 and the arbitrage opportunity disappears.
A similar arbitrage opportunity arises if the bond price is higher than $952.38. For example, suppose the
bond is trading for $960. In that case, we should sell the bond and invest $952.38 at the bank. As shown
in Table 3.4, we then earn $7.62 in cash today, yet keep our future cash flows unchanged by replacing
the $1000 we would have received from the bond with the $1000 we will receive from the bank. Once
again, as people begin sell- ing the bond to exploit this opportunity, the price will fall until it reaches
$952.38 and the arbitrage opportunity disappears.
When the bond is overpriced, the arbitrage strategy involves selling the bond and invest- ing some of the
proceeds. But if the strategy involves selling the bond, does this mean that only the current owners of the
bond can exploit it? The answer is no; in financial markets it is possible to sell a security you do not own
by doing a short sale. In a short sale, the person who intends to sell the security first borrows it from
someone who already owns it. Later, that person must either return the security by buying it back or pay
the owner the cash flows he or she would have received. For example, we could short sell the bond in the
example effectively promising to repay the current owner $1000 in one year. By executing a short sale, it
is possible to exploit the arbitrage opportunity when the bond is overpriced even if you do not own it.
TABLE 3.4
Sell the bond
Invest at the bank Net cash flow
Net Cash Flows from Selling the Bond and Investing
Today ($)
+960.00 -952.38 +7.62
In One Year ($)
-1000.00 +1000.00 0.00
74
Chapter 3 Financial Decision Making and the Law of One Price EXAMPLE 3.6
Computing the No-Arbitrage Price
Problem
Consider a security that pays its owner $100 today and $100 in one year, without any risk. Sup- pose the risk-free interest rate is 10%.
What is the no-arbitrage price of the security today (before the first $100 is paid)? If the security is trading for $195, what arbitrage
opportunity is available?
Solution
We need to compute the present value of the security’s cash flows. In this case there are two cash flows: $100 today, which is already
in present value terms, and $100 in one year. The present value of the second cash flow is
$100 in one year , (1.10 $ in one year/$ today) = $90.91 today
Therefore, the total present value of the cash flows is $100 + $90.91 = $190.91 today, which is the no-arbitrage price of the security.
If the security is trading for $195, we can exploit its overpricing by selling it for $195. We can then use $100 of the sale proceeds to
replace the $100 we would have received from the security today and invest $90.91 of the sale proceeds at 10% to replace the $100
we would have received in one year. The remaining $195 – $100 – $90.91 = $4.09 is an arbitrage profit.
Determining the No-Arbitrage Price. We have shown that at any price other than $952.38, an arbitrage
opportunity exists for our bond. Thus, in a normal market, the price of this bond must be $952.38. We
call this price the no-arbitrage price for the bond.
By applying the reasoning for pricing the simple bond, we can outline a general process for pricing other
securities:
1. Identify the cash flows that will be paid by the security.
2. Determine the “do-it-yourself ” cost of replicating those cash flows on our own; that is, the present
value of the security’s cash flows.
Unless the price of the security equals this present value, there is an arbitrage opportunity. Thus, the
general formula is
No-Arbitrage Price of a Security
Price(Security) = PV (All cash flows paid by the security) (3.3)
Determining the Interest Rate from Bond Prices. Given the risk-free interest rate, the no-arbitrage price of a
risk-free bond is determined by Eq. 3.3. The reverse is also true: If we know the price of a risk-free bond,
we can use Eq. 3.3 to determine what the risk-free interest rate must be if there are no arbitrage
opportunities.
For example, suppose a risk-free bond that pays $1000 in one year is currently trading with a competitive
market price of $929.80 today. From Eq. 3.3, we know that the bond’s price equals the present value of
the $1000 cash flow it will pay:
$929.80 today = ($1000 in one year) , (1 + rf ) We can rearrange this equation to determine the risk-free
interest rate:
1 + rf =
$1000 in one year
= 1.0755 $ in one year/$ today $929.80 today
3.5 No-Arbitrage and Security Prices 75
That is, if there are no arbitrage opportunities, the risk-free interest rate must be 7.55%. Interest rates are
calculated by this method in practice. Financial news services report current interest rates by deriving
these rates based on the current prices of risk-free government bonds trading in the market.
you earn from invest-
Note that the risk-free interest rate equals the percentage gain that
ing in the bond, which is called the bond’s return:
Return =
Gain at End of Year
Initial Cost
=
1000 – 929.80
=
1000
– 1 = 7.55% (3.4) 929.80 929.80
Thus, if there is no arbitrage, the risk-free interest rate is equal to the return from invest- ing in a risk-free
bond. If the bond offered a higher return than the risk-free interest rate, then investors would earn a profit
by borrowing at the risk-free interest rate and investing in the bond. If the bond had a lower return than
the risk-free interest rate, investors would sell the bond and invest the proceeds at the risk-free interest
rate. No arbitrage is therefore equivalent to the idea that all risk-free investments should offer investors
the same return.
The NPV of Trading Securities and Firm Decision Making
We have established that positive-NPV decisions increase the wealth of the firm and its investors. Think
of buying a security as an investment decision. The cost of the decision is the price we pay for the
security, and the benefit is the cash flows that we will receive from owning the security. When securities
trade at no-arbitrage prices, what can we conclude about the value of trading them? From Eq. 3.3, the
cost and benefit are equal in a normal market and so the NPV of buying a security is zero:
NPV (Buy security) = PV (All cash flows paid by the security) – Price (Security) =0
Similarly, if we sell a security, the price we receive is the benefit and the cost is the cash flows we give
up. Again the NPV is zero:
NPV (Sell security) = Price (Security) – PV (All cash flows paid by the security) =0
Thus, the NPV of trading a security in a normal market is zero. This result is not sur- prising. If the NPV
of buying a security were positive, then buying the security would be equivalent to receiving cash
today—that is, it would present an arbitrage opportunity. Because arbitrage opportunities do not exist in
normal markets, the NPV of all security trades must be zero.
Another way to understand this result is to remember that every trade has both a buyer and a seller. In a
competitive market, if a trade offers a positive NPV to one party, it must give a negative NPV to the
other party. But then one of the two parties would not agree to the trade. Because all trades are voluntary,
they must occur at prices at which neither party is losing value, and therefore for which the trade is zero
NPV.
The insight that security trading in a normal market is a zero-NPV transaction is a critical building block
in our study of corporate finance. Trading securities in a normal market neither creates nor destroys
value: Instead, value is created by the real investment projects in which the firm engages, such as
developing new products, opening new stores,
76
Chapter 3 Financial Decision Making and the Law of One Price
or creating more efficient production methods. Financial transactions are not sources of value but instead
serve to adjust the timing and risk of the cash flows to best suit the needs of the firm or its investors.
An important consequence of this result is the idea that we can evaluate a decision by focusing on its real
components, rather than its financial ones. That is, we can separate the firm’s investment decision from
its financing choice. We refer to this concept as the Sepa- ration Principle:
Security transactions in a normal market neither create nor destroy value on their own. Therefore, we
can evaluate the NPV of an investment decision separately from the decision the firm makes regarding
how to finance the investment or any other security transactions the firm is considering.
Separating Investment and Financing
Problem
Your firm is considering a project that will require an upfront investment of $10 million today and will produce $12 million
flow for the firm in one year without risk. Rather than pay for the $10 million investment entirely using its own cash, the fir
considering raising additional funds by issuing a security that will pay investors $5.5 million in one year. Suppose the risk-f
rate is 10%. Is pursuing this project a good decision without issuing the new security? Is it a good decision with the new sec
Solution
Without the new security, the cost of the project is $10 million today and the benefit is $12 mil- lion in one year. Converting
benefit to a present value
$12 million in one year , (1.10 $ in one year/$ today) = $10.91 million today
we see that the project has an NPV of $10.91 million – $10 million = $0.91 million today. Now suppose the firm issues the n
security. In a normal market, the price of this security
will be the present value of its future cash flow:
Price(Security) = $5.5 million , 1.10 = $5 million today
Thus, after it raises $5 million by issuing the new security, the firm will only need to invest an additional $5 million to take
To compute the project’s NPV in this case, note that in one year the firm will receive the $12 million payout of the project, b
$5.5 million to the investors in the new security, leaving $6.5 million for the firm. This amount has a present value of
$6.5 million in one year , (1.10 $ in one year/$ today) = $5.91 million today
Thus, the project has an NPV of $5.91 million – $5 million = $0.91 million today, as before. In either case, we get the sam
the NPV. The separation principle indicates that we will get the same result for any choice of financing for the firm that occ
normal market. We can therefore evaluate the project without explicitly considering the different financing possibilities the firm might choose.
EXAMPLE 3.7
Valuing a Portfolio
So far, we have discussed the no-arbitrage price for individual securities. The Law of One Price also has
implications for packages of securities. Consider two securities, A and B.
3.5 No-Arbitrage and Security Prices 77
Stock Index Arbitrage
Value additivity is the principle behind a type of trading activity known as stock index arbitrage. Common stock indices (such
Dow Jones Industrial Average and the Standard and Poor’s 500) represent portfolios of individual stocks. It is possible to trad
individual stocks in an index on the New York Stock Exchange and NASDAQ. It is also possible to trade the entire index (as
security) on the futures exchanges in Chicago, or as an exchange-traded fund (ETF) on the NYSE. When the price of the inde
is below the total price of the individual stocks, traders buy the index and sell the stocks to capture the price difference. Simil
the price of the index security is above the
total price of the individual stocks, traders sell the index and buy the individual stocks. The investment banks that engage in s
arbitrage automate the process by tracking the prices and submitting the orders via computer; as a result, this activity is also r
as “program trading.” It is not uncommon for 20% to 30% of the daily volume of trade on the NYSE to be due to index arbitr
activity via program trading.* The actions of these arbitrageurs ensure that the prices of the index securities and the individua
prices track each other very closely.
*See http://usequities.nyx.com/markets/program-trading
Suppose a third security, C, has the same cash flows as A and B combined. In this case, security C is
equivalent to a combination of the securities A and B. We use the term portfolio to describe a collection
of securities. What can we conclude about the price of security C as compared to the prices of A and B?
Value Additivity. Because security C is equivalent to the portfolio of A and B, by the Law of One Price,
they must have the same price. This idea leads to the relationship known as value additivity; that is, the
price of C must equal the price of the portfolio, which is the combined price of A and B:
Value Additivity
Price(C) = Price(A + B) = Price(A) + Price(B) (3.5)
Because security C has cash flows equal to the sum of A and B, its value or price must be the sum of the
values of A and B. Otherwise, an obvious arbitrage opportunity would exist. For example, if the total
price of A and B were lower than the price of C, then we could make a profit buying A and B and selling
C. This arbitrage activity would quickly push prices until the price of security C equals the total price of
A and B.
EXAMPLE 3.8
Valuing an Asset in a Portfolio
Problem
Holbrook Holdings is a publicly traded company with only two assets: It owns 60% of Harry’s Hotcakes restaurant chain
and an ice hockey team. Suppose the market value of Holbrook Hold- ings is $160 million, and the market value of the
entire Harry’s Hotcakes chain (which is also publicly traded) is $120 million. What is the market value of the hockey
team?
Solution
We can think of Holbrook as a portfolio consisting of a 60% stake in Harry’s Hotcakes and the hockey team. By value
additivity, the sum of the value of the stake in Harry’s Hotcakes and the hockey team must equal the $160 million market
value of Holbrook. Because the 60% stake in Harry’s Hotcakes is worth 60% * $120 million = $72 million, the hockey
team has a value of $160 million – $72 million = $88 million.
78 Chapter 3 Financial Decision Making and the Law of One Price
GLOBAL FINANCIAL CRISIS
In the first half of 2008, as the extent and severity of the decline in the housing market became apparent, investors
became increasingly worried about the value of securities that were backed by residential home mortgages. As a result,
the volume of trade in the multi-trillion dollar market for mortgage-backed securities plummeted over 80% by August
2008. Over the next two months, trading in many of these securities ceased altogether, making the markets for these
securities increasingly illiquid.
Competitive markets depend upon liquidity—there must be sufficient buyers and sellers of a security so that it is possible
to trade at any time at the current market price. When markets become illiquid it may not be possible to trade at the
posted price. As a consequence, we can no lon- ger rely on market prices as a measure of value.
The collapse of the mortgage-backed securities market created two problems. First was the loss of trading oppor- tunities,
making it difficult for holders of these securities to sell them. But a potentially more significant problem was the loss of
information. Without a liquid, competitive market for these securities, it became impossible to reliably
value these securities. In addition, given that the value of the banks holding these securities was based on the sum of all
projects and investments within them, investors could not value the banks either. Investors reacted to this uncertainty by
selling both the mortgage-backed securities and securities of banks that held mortgage-backed securities. These actions
further compounded the problem by driving down prices to seemingly unrealistically low levels and thereby threatening
the solvency of the entire financial system.
The loss of information precipitated by the loss of liquid- ity played a key role in the breakdown of credit markets. As
both investors and government regulators found it increas- ingly difficult to assess the solvency of the banks, banks
found it difficult to raise new funds on their own and also shied away from lending to other banks because of their
concerns about the financial viability of their competitors. The result was a breakdown in lending. Ultimately, the government was forced to step in and spend hundreds of bil- lions of dollars in order to (1) provide new capital to support the
banks and (2) provide liquidity by creating a market for the now “toxic” mortgage-backed securities.
Liquidity and the Informational Role of Prices
More generally, value additivity implies that the value of a portfolio is equal to the sum of the values of
its parts. That is, the “à la carte” price and the package price must coincide.7
Value Additivity and Firm Value. Value additivity has an important consequence for the value of an entire
firm. The cash flows of the firm are equal to the total cash flows of all projects and investments within
the firm. Therefore, by value additivity, the price or value of the entire firm is equal to the sum of the
values of all projects and investments within it. In other words, our NPV decision rule coincides with
maximizing the value of the entire firm:
To maximize the value of the entire firm, managers should make decisions that maximize NPV. The NPV
of the decision represents its contribution to the overall value of the firm.
Where Do We Go from Here?
The key concepts we have developed in this chapter—the Valuation Principle, Net Present Value, and
the Law of One Price—provide the foundation for financial decision making.
7
This feature of financial markets does not hold in many other noncompetitive markets. For example, a round-trip airline ticket
often costs much less than two separate one-way tickets. Of course, airline tickets are not sold in a competitive market—you cannot
buy and sell the tickets at the listed prices. Only airlines can sell tickets, and they have strict rules against reselling tickets.
Otherwise, you could make money buy- ing round-trip tickets and selling them to people who need one-way tickets.
CONCEPT CHECK
The Law of One Price allows us to determine the value of stocks, bonds, and other securities, based on their cash flows, and
validates the optimality of the NPV decision rule in identify- ing projects and investments that create value. In the remainder of
the text, we will build on this foundation and explore the details of applying these principles in practice.
For simplicity, we have focused in this chapter on projects that were not risky, and thus had known costs and benefits. The same
fundamental tools of the Valuation Principle and the Law of One Price can be applied to analyze risky investments as well, and
we will look in detail at methods to assess and value risk in Part IV of the text. Those seeking some early insights and key
foundations for this topic, however, are strongly encouraged to read the appendix to this chapter. There we introduce the idea
that investors are risk averse, and then use the principle of no-arbitrage developed in this chapter to demonstrate two fundamental insights regarding the impact of risk on valuation:
1. When cash flows are risky, we must discount them at a rate equal to the risk-free interest rate plus an appropriate risk
premium; and,
2. Theappropriateriskpremiumwillbehigherthemoretheproject’sreturnstendto vary with the overall risk in the economy.
Finally, the chapter appendix also addresses the important practical issue of transactions costs. There we show that when
purchase and sale prices, or borrowing and lending rates differ, the Law of One Price will continue to hold, but only up to the
level of transactions costs.

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