Transcript Chapter 3

Chapter 3
Arbitrage and Financial Decision Making
Chapter Outline
3.1 Valuing Costs and Benefits
3.2 Interest Rates and the Time Value of Money
3.3 Present Value and the NPV Decision Rule
3.4 Arbitrage and the Law of One Price
3.5 No-Arbitrage and Security Prices
3.6 The Price of Risk
3.7 Arbitrage with Transaction Costs
2
Learning Objectives
1.
2.
Assess the relative merits of two-period projects using
net present value.
Define the term “competitive market,” give examples of
markets that are competitive and some that aren’t, and
discuss the importance of a competitive market in
determining the value of a good.
3.
Explain why maximizing NPV is always the correct
decision rule.
4.
Define arbitrage, and discuss its role in asset pricing.
How does it relate to the Law of One Price?
5.
Calculate the no-arbitrage price of an investment
opportunity.
3
Learning Objectives (cont'd)
6.
Show how value additivity can be used to help
managers maximize the value of the firm.
7.
Describe the Separation Principle.
8.
Calculate the value of a risky asset, using the Law
of One Price.
9.
10.
Describe the relationship between a security’s
risk premium and its correlation with returns of
other securities.
Describe the effect of transactions costs on
arbitrage and the Law of One Price.
4
3.1 Valuing Costs and Benefits

Identify Costs and Benefits

May need help from other areas in identifying the
relevant costs and benefits

Marketing

Economics

Organizational Behavior

Strategy

Operations
5
Using Market Prices to Determine Cash Values

Suppose a jewelry manufacturer has the
opportunity to trade 10 ounces of platinum
and receive 20 ounces of gold today. To
compare the costs and benefits, we first need
to convert them to a common unit.
6
Using Market Prices to Determine Cash Values
(cont'd)

Suppose gold can be bought and sold for a
current market price of $250 per ounce. Then
the 20 ounces of gold we receive has a cash
value of:

(20 ounces of gold)  ($250/ounce) = $5000 today

7
Using Market Prices to Determine Cash Values
(cont'd)

Similarly, if the current market price for
platinum is $550 per ounce, then the 10
ounces of platinum we give up has a cash
value of:

(10 ounces of platinum)  ($550/ounce) = $5500

8
Using Market Prices to Determine Cash Values
(cont'd)

Therefore, the jeweler’s opportunity has a
benefit of $5000 today and a cost of $5500
today. In this case, the net value of the
project today is:


$5000 – $5500 = –$500
Because it is negative, the costs exceed the
benefits and the jeweler should reject the
trade.
9
Example 3.1
10
Example 3.1 (cont'd)
11
Example 3.2
12
Example 3.2 (cont'd)
13
When Competitive Market Prices Are Not
Available

When competitive prices are not available,
prices may be one sided. For example, at
retail stores you can buy at the posted price,
but you cannot sell the good to the store at
that same price. One-sided prices determine
the maximum value of the good (since it can
always be purchased at that price), but an
individual may value it for much less
depending on his or her preferences for the
good.
14
Example 3.3
15
Example 3.3 (cont'd)
16
3.2 Interest Rates and the Time Value of
Money

Time Value of Money


Consider an investment opportunity with the
following certain cash flows.

Cost: $100,000 today

Benefit: $105,000 in one year
The difference in value between money today and
money in the future is due to the time value of
money.
17
The Interest Rate: An Exchange Rate Across Time

The rate at which we can exchange money
today for money in the future is determined
by the current interest rate.

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.

Risk–Free Interest Rate (Discount Rate), rf: The interest
rate at which money can be borrowed or lent without risk.

Interest Rate Factor = 1 + rf

Discount Factor = 1 / (1 + rf)
18
Example 3.4
19
Example 3.4 (cont'd)
20
Alternative Example 3.4

Problem

The cost of replacing a fleet of company trucks
with more energy efficient vehicles was $100
million
in 2006.

The cost is estimated to rise by 8.5% in 2007.

If the interest rate was 4%, what was the cost
of a delay in terms of dollars in 2007?
21
Alternative Example 3.4

Solution

If the project were delayed, it’s cost in 2007 would
be:


Compare this amount to the cost of $100 million in
2006 using the interest rate of 4%:


$100 million × (1.085) = $108.5 million
$108.5 million ÷ 1.04 = $104.33 million in 2006 dollars.
The cost of a delay of one year would be:

$104.33 million – $100 million = $4.33 million in 2006
dollars.
22
Figure 3.1 Converting Between Dollars Today and
Gold, Euros, or Dollars in the Future
23
3.3 Present Value and the NPV Decision Rule

The net present value (NPV) of a project or
investment is the difference between the
present value of its benefits and the present
value of
its costs. NPV  PV (Benefits)  PV (Costs)

Net Present Value
NPV  PV (All project cash flows)
24
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.
25
The NPV Decision Rule (cont'd)

Accepting or Rejecting a Project

Accept those projects with positive NPV because
accepting them is equivalent to receiving their
NPV in cash today.

Reject those projects with negative NPV because
accepting them would reduce the wealth of
investors.
26
Example 3.5
27
Example 3.5 (cont'd)
28
Choosing Among Projects
29
Choosing Among Projects (cont'd)

All three projects have positive NPV, and we
would accept all three if possible.

If we must choose only one project, Project B
has the highest NPV and therefore is the best
choice.
30
NPV and Individual Preferences

Although Project B has the highest NPV,
what if we do not want to spend the $20 for
the cash outlay? Would Project A be a better
choice? Should this affect our choice of
projects?

NO! As long as we are able to borrow and
lend at the risk-free interest rate, Project B is
superior whatever our preferences regarding
the timing of the cash flows.
31
NPV and Individual Preferences (cont'd)
32
NPV and Individual Preferences (cont'd)

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.
33
Figure 3.2
Comparing
Projects
A, B, & C
34
3.4 Arbitrage and the Law of One Price

Arbitrage


The practice of buying and selling equivalent
goods in different markets to take advantage of a
price difference. An arbitrage opportunity occurs
when it is possible to make a profit without taking
any risk or making any investment.
Normal Market

A competitive market in which there are no
arbitrage opportunities.
35
3.4 Arbitrage and the Law of One Price
(cont'd)

Law of One Price

If equivalent investment opportunities trade
simultaneously in different competitive markets,
then they must trade for the same price in both
markets.
36
3.5 No-Arbitrage and Security Prices

Valuing a Security

Assume a security promises a risk-free payment
of $1000 in one year. If the risk-free interest rate is
5%, what can we conclude about the price of this
bond in a normal market?
PV ($1000 in one year)  ($1000 in one year)  (1.05 $ in one year / $ today)
 $952.38 today

Price(Bond) = $952.38
37
3.5 No-Arbitrage and Security Prices (cont'd)

Valuing a Security (cont’d)

What if the price of the bond is not $952.38?


Assume the price is $940.
The opportunity for arbitrage will force the price of
the bond to rise until it is equal to $952.38.
38
3.5 No-Arbitrage and Security Prices (cont'd)

Valuing a Security (cont’d)

What if the price of the bond is not $952.38?


Assume the price is $960.
The opportunity for arbitrage will force the price of
the bond to fall until it is equal to $952.38.
39
Determining the No-Arbitrage Price

Unless the price of the security equals the
present value of the security’s cash flows, an
arbitrage opportunity will appear.

No Arbitrage Price of a Security
Price(Security)  PV (All cash flows paid by the security)
40
Example 3.6
41
Example 3.6 (cont'd)
42
Determining the Interest Rate From Bond
Prices

If we know the price of a risk-free bond, we
can use
Price(Security)  PV (All cash flows paid by the security)
to determine what the risk-free interest rate
must be if there are no arbitrage opportunities.
43
Determining the Interest Rate From Bond
Prices (cont'd)

Suppose a risk-free bond that pays $1000 in
one year is currently trading with a
competitive market price of $929.80 today.
The bond’s price must equal the present
value of the $1000 cash flow it will pay.
44
Determining the Interest Rate From Bond
Prices (cont'd)
$929.80 today  ($1000 in one year)  (1  rf $ in one year / $ today)
1  rf 

$1000 in one year
 1.0755 $ in one year / $ today
$929.80 today
The risk-free interest rate must be 7.55%.
45
The NPV of Trading Securities

In a normal market, the NPV of buying or
selling a security is zero.
NPV (Buy security)  PV (All cash flows paid by the security)  Price(Security)
 0
NPV (Sell security)  Price(Security)  PV (All cash flows paid by the security)
 0
46
The NPV of Trading Securities (cont’d)

Separation Principle

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.
47
Example 3.7
48
Example 3.7 (cont'd)
49
Valuing a Portfolio

The Law of One Price also has implications
for packages of securities.

Consider two securities, A and B. Suppose a third
security, C, has the same cash flows as A and B
combined. In this case, security C is equivalent to
a portfolio, or combination, of the securities A and
B.
Price(C)  Price(A  B)  Price(A)  Price(B)

Value Additivity
50
Example 3.8
51
Example 3.8 (cont'd)
52
Alternative Example 3.8

Problem


Moon Holdings is a publicly traded company with
only three assets:

It owns 50% of Due Beverage Co., 70% of Mountain
Industries, and 100% of the Oxford Bears, a football
team.

The total market value of Moon Holdings is $200 million,
the total market value of Due Beverage Co. is $75
million and the total market value of Mountain Industries
is $100 million.
What is the market value of the Oxford Bears?
53
Alternative Example 3.8

Solution

Think of Moon as a portfolio consisting of a:

50% stake in Due Beverage


70% stake in Mountain Industries



50% × $75 million = $37.5 million
70% × $100 million = $70 million
100% stake in Oxford Bears
Under the Value Added Method, the sum of the
value of the stakes in all three investments must
equal the $200 million market value of Moon.

The Oxford Bears must be worth:

$200 million − $37.5 million − $70 million = $92.5 million
54
3.6 The Price of Risk

Risky Versus Risk-free Cash Flows

Assume there is an equal probability of either a
weak economy or strong economy.
55
3.6 The Price of Risk (cont'd)

Risky Versus Risk-free Cash Flows (cont’d)
Price(Risk-free Bond)  PV(Cash Flows)
 ($1100 in one year)  (1.04 $ in one year / $ today)
 $1058 today

Expected Cash Flow (Market Index)

½ ($800) + ½ ($1400) = $1100

Although both investments have the same expected
value, the market index has a lower value since it has a
greater amount of risk.
56
Risk Aversion and the Risk Premium

Risk Aversion


Investors prefer to have a safe income rather than
a risky one of the same average amount.
Risk Premium


The additional return that investors expect to earn
to compensate them for a security’s risk.
When a cash flow is risky, to compute its present
value we must discount the cash flow we expect
on average at a rate that equals the risk-free
interest rate plus an appropriate risk premium.
57
Risk Aversion and the Risk Premium (cont’d)
Expected return of a risky investment 

Market return if the economy is strong


(1400 – 1000) / 1000 = 40%
Market return if the economy is weak


Expected Gain at end of year
Initial Cost
(800 – 1000) / 1000 = –20%
Expected market return

½ (40%) + ½ (–20%) = 10%
58
The No-Arbitrage Price of a Risky Security

If we combine security A with a risk-free bond that pays
$800 in one year, the cash flows of the portfolio in one year
are identical to the cash flows of the market index.

By the Law of One Price, the total market value of the bond
and security A must equal $1000, the value of the market
index.
59
The No-Arbitrage Price of a Risky Security
(cont'd)

Given a risk-free interest rate of 4%, the market
price of the bond is:

($800 in one year) / (1.04 $ in one year/$ today) = $769
today

Therefore, the initial market price of security A is
$1000 – $769 = $231.
60
Risk Premiums Depend on Risk

If an investment has much more variable
returns, it must pay investors a higher risk
premium.
61
Risk Is Relative to the Overall Market

The risk of a security must be evaluated in
relation to the fluctuations of other
investments in the economy.

A security’s risk premium will be higher the
more its returns tend to vary with the overall
economy and the market index.

If the security’s returns vary in the opposite
direction of the market index, it offers
insurance and will have a negative risk
premium.
62
Risk Is Relative to the Overall Market (cont'd)
63
Example 3.9
64
Example 3.9 (cont'd)
65
Risk, Return, and Market Prices

When cash flows are risky, we can use the
Law of One Price to compute present values
by constructing a portfolio that produces cash
flows with identical risk.
66
Figure 3.3 Converting Between Dollars
Today and Dollars in One Year with Risk

Computing prices in this way is equivalent to
converting between cash flows today and the
expected cash flows received in the future using a
discount rate rs that includes a risk premium
appropriate for the investment’s risk:
rs  rf  ( risk premium for investment s)
67
Example 3.10
68
Example 3.10 (cont'd)
69
3.7 Arbitrage with Transactions Costs

What consequence do transaction costs have
for no-arbitrage prices and the Law of One
Price?

When there are transactions costs, arbitrage
keeps prices of equivalent goods and securities
close to each other. Prices can deviate, but not by
more than the transactions cost of the arbitrage.
70
Example 3.11
71
Example 3.11 (cont'd)
72