chapter 3 arbitrage and financial decision making -...

Copyright © 2011 Pearson Prentice Hall. All rights reserved.

Chapter 3

Arbitrage and Financial Decision Making

Copyright © 2011 Pearson Prentice Hall. All rights reserved. 3-2

Chapter Outline

3.1 Valuing Decisions

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


3.1 Valuing Decisions

• Identify Costs and Benefits – May need help from other areas in identifying

the relevant costs and benefits • Marketing • Economics • Organizational Behavior • Strategy • Operations


Using Market Prices to Determine Cash Values

• Competitive Market – A market in which goods can be bought and

sold at the same price. • In evaluating the jeweler’s decision, we

used the current market price to convert from ounces of platinum 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.


Textbook Example 3.1


Textbook Example 3.1 (cont'd)


Alternative Example 3.1

• Problem – Your car recently broke down and it needs

$2,000 in repairs. But today is your lucky day because you have just won a contest where the prize is either a new motorcycle, with a MSRP of $15,000, or $10,000 in cash. You do not have a motorcycle license, nor do you plan on getting one. You estimate you could sell the motorcycle for $12,000. Which prize should you choose?


Alternative Example 3.1 (cont'd)

• Solution – Competitive markets, not your personal

preferences (or the MSRP of the motorcycle), are relevant here: One Motorcycle with a market value of $12,000 or $10,000 cash. Instead of taking the cash, you should accept the motorcycle, sell it for $12,000, use $2,000 to pay for your car repairs, and still have $10,000 left over.


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.


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)


The Interest Rate: An Exchange Rate Across Time (cont'd)

• Value of Investment in One Year – 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



• Value of Investment 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.


The Interest Rate: An Exchange Rate Across Time (cont'd) • Value of Investment Today

– Consider the benefit of $105,000 in one year. What is the equivalent amount in terms of dollars today? Benefit = ($105,000 in one year) ÷ (1.07 $ in one

year/$ today)

= ($105,000 in one year) × 1/1.07 = $98,130.84 today

– This is the amount the bank would lend to us today if we promised to repay $105,000 in one year.



• Value of Investment Today – 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.



• Present Versus Future Value – This demonstrates that our decision is the same

whether we express the value of the investment in terms of dollars in one year or dollars today. If we convert from dollars today to dollars in one year,

(−$1869.16 today) × (1.07 $ in one year/$ today) = −$2000 in one year.

– The two results are equivalent, but expressed as values at different points in time.



• Present Versus Future Value – 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 of the investment.



• Discount Factors and Rate – We can interpret

1 1 0.934581 1.07r

= =+

11 r+as the price today of $1 in one year. The amount is

called the one- year discount factor. The risk-free rate is also referred to as the discount rate for a risk-free investment.



• Problem – The cost of replacing a fleet of company trucks

with more energy efficient vehicles was $100 million in 2009.

– The cost is estimated to rise by 8.5% in 2010.

– If the interest rate was 4%, what was the cost of a delay in terms of dollars in 2009?



• Solution – If the project were delayed, its cost in 2010

would be: • $100 million × (1.085) = $108.5 million

– Compare this amount to the cost of $100 million in 2009 using the interest rate of 4%: • $108.5 million ÷ 1.04 = $104.33 million in 2009

dollars.

– The cost of a delay of one year would be: • $104.33 million – $100 million = $4.33 million in

2009 dollars.


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. – Net Present Value

(Benefits) (Costs)= −NPV PV PV(All project cash flows)=NPV PV


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.


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.


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 which will lead to the largest increase in the value of the firm.


Choosing Among Alternatives (cont'd)

Table 3.1 Cash Flows and NPVs for Web Site Business Alternatives


NPV and Cash Needs

• 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.


NPV and Cash Needs (cont'd)

Table 3.2 Cash Flows of Hiring and Borrowing Versus Selling and Investing


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.


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.


3.5 No-Arbitrage and Security Prices

• Valuing a Security with the Law of One Price – 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?

• Price(Bond) = $952.38

($1000 in one year) ($1000 in one year) (1.05 $ in one year / $ today)$952.38 today

= ÷=

PV


Identifying Arbitrage Opportunities with Securities

• 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.

Table 3.3 Net Cash Flows from Buying the Bond and Borrowing


Identifying Arbitrage Opportunities with Securities

• 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.

Table 3.4 Net Cash Flows from Selling the Bond and Investing


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) (All cash flows paid by the security)= PV


Determining the Interest Rate From Bond Prices

• If we know the price of a risk-free bond, we can use

to determine what the risk-free interest rate must be if there are no arbitrage opportunities.

Price(Security) (All cash flows paid by the security)= PV


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.


Determining the Interest Rate From Bond Prices (cont'd)

• The risk-free interest rate must be 7.55%.

$929.80 today ($1000 in one year) (1 $ in one year / $ today)= ÷ + fr

$1000 in one year1 1.0755 $ in one year / $ today$929.80 today

+ = =fr


The NPV of Trading Securities and Firm Decision Making

• In a normal market, the NPV of buying or selling a security is zero.

(Buy security) (All cash flows paid by the security) Price(Security) 0

= −=

NPV PV

(Sell security) Price(Security) (All cash flows paid by the security) 0

= −=

NPV PV


The NPV of Trading Securities and Firm Decision Making (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.


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.

• Value Additivity

Price(C) Price(A B) Price(A) Price(B)= + = +



• 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?


Alternative Example 3.8 (cont'd)

• Solution – Think of Moon as a portfolio consisting of a:

• 50% stake in Due Beverage – 50% × $75 million = $37.5 million

• 70% stake in Mountain Industries – 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


Appendix: The Price of Risk

• Risky Versus Risk-free Cash Flows

– Assume there is an equal probability of either a weak economy or strong economy.

Table 3A.1 Cash Flows and Market Prices (in $) of a Risk-Free Bond and an Investment in the Market Portfolio


Textbook Example 3A.2


Textbook Example 3A.2 (cont'd)


Alternative Example 3A.2

• Problem – Consider a risky stock with a cash flow of

$1500 when the economy is strong and $800 when the economy is weak. Each state of the economy has an equal probability of occurring. Suppose an 8% risk premium is appropriate for this particular stock. If the risk-free interest rate is 2%, what is the price of the stock today?


Alternative Example 3A.2 (cont’d)

• Solution – From Eq. 3A.2, the appropriate discount rate

for the stock is:

rs = rf + (Risk Premium for the Stock) = 2% + 8% = 10%

– The expected cash flow of the bond is

½($1,500) + ½($800) = $1,150 in one year. Thus, the price of the stock today is $1,150/1.10 = $1,045.45.


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.


Textbook Example 3A.3

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