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Chapter 3

1. Assess the relative merits of two-period projects using net present value.

2. 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.

6. Show how value additivity can be used to help managers maximize the value of the firm.

7. Describe the Separation Principle.

Identify Costs and Benefits May need help from other areas in identifying the relevant costs and benefits Marketing

Economics

Organizational Behavior

Strategy

Operations

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Suppose a jewelry manufacturer has the opportunity to trade 10 ounces of gold and receive 20 ounces of palladium today. To compare the costs and benefits, we first need to convert them to a common unit.

Suppose gold can be bought and sold for a current market price of $1,500 per ounce. Then the 10 ounces of gold we give up has a cash value of:

(10 ounces of gold) X ($1,500/ounce) = $15,000 today

Similarly, if the current market price for palladium is $600 per ounce, then the 20 ounces of palladium we receive has a cash value of:

(20 ounces of palladium) X ($600/ounce) = $12,000

Therefore, the jeweler’s opportunity has a benefit of $12,000 today and a cost of $15,000 today. In this case, the net value of the project today is:

$12,000 – $15,000 = –$3,000

Because it is negative, the costs exceed the benefits and the jeweler should reject the trade.

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.

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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.

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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)

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.

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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.

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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 (FV) of the investment.

Discount Factors and Rate

We can interpret as the price today of $1 in one year.

The amount 1/(1 + r) is called the one- year discount factor.

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

11

11.07

0.93458

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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

Allprojectcashflows

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

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.

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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.

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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.

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.

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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.

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

$1000inoneyear $1000inoneyear 1.05$inoneyear/$today$952.38today

What if the price of the bond is not $952.38? Assume the price is $940.

You make $12.38 with no net investment and no risk (an arbitrage opportunity)

The opportunity for arbitrage will force the price of the bond to rise until it is equal to $952.38.

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What if the price of the bond is not $952.38? Assume the price is $960.

You make $7.63 with no net investment and no risk (an arbitrage opportunity)

The opportunity for arbitrage will force the price of the bond to fall until it is equal to $952.38.

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 Allcashflowspaidbythesecurity

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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 Allcashflowspaidbythesecurity

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.

The risk-free interest rate must be 7.55%.

$929.80today $1000inoneyear 1 $inoneyear/$today

1 $1000inoneyear$929.80today

1.0755$inoneyear/$today

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

If not, there is an arbitrage opportunity

Buysecurity Allcashflowspaidbythesecurity Price Security0

Sellsecurity Price Security Allcashflowspaidbythesecurity0

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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.

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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?

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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

Value Additivity and Firm Value

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.

Impact of Risk on Valuation

When cash flows are risky, we must discount them at a rate equal to the risk-free interest rate plus an appropriate risk premium.

The appropriate risk premium will be higher the more the project’s returns tend to vary with overall risk in the economy.

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If a Big Mac at McDonald’s costs $4.00 in the United States but only $3.75 (U.S. dollars) in Canada, does this violate the Law of One Price? Is there an arbitrage opportunity? Why or why not?

1. If gasoline trades in a competitive market, would a transportation company that has a use for the gasoline value it differently than another investor?

2. How do you compare benefits at different points in time?

3. If interest rates fall, what happens to the value today of a promise of money in one year?

4. What is the NPV decision rule?

5. Does the NPV decision rule depend on the investor’s preferences?

6. What is the Law of One Price?

7. What is Arbitrage?

8. If a firm makes an investment that has a negative NPV, how does the value of the firm change?

9. What is the Separation Principle?

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Chapter 3

1. Calculate the value of a risky asset, using the Law of One Price.

2. Describe the relationship between a security’s risk premium and its correlation with returns of other securities.

3. Describe the effect of transactions costs on arbitrage and the Law of One Price.

Risky Versus Risk-free Cash Flows

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

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Risky Versus Risk-free Cash Flows (cont’d)

Expected Cash Flow (Market Index) ½ ($800) + ½ ($1400) = $1100

The price of the market index today is $1000

Although both investments have the same expected value in 1 year, the market index has a lower value since it has a greater amount of risk.

The expected rate of return on the market index is 10%

Price Risk freeBond PV CashFlows $1100inoneyear 1.04$inoneyear/$today$1058today

$1000today $1100inoneyear 1 $inoneyear/$today

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. IN the example, the risk premium is 6%

Market return if the economy is strong (1400 – 1000) / 1000 = 40%

Market return if the economy is weak (800 – 1000) / 1000 = –20%

Expected market return ½ (40%) + ½ (–20%) = 10%

Expectedreturnofariskyinvestment ExpectedGainatendofyear

InitialCost

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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.

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.

If an investment has much more variable returns, it must pay investors a higher risk premium.

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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.

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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.

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.