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

Learning Objectives 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. Explain why maximizing NPV is always the correct decision rule. Define arbitrage, and discuss its role in asset pricing. How does it relate to the Law of One Price? Calculate the no-arbitrage price of an investment opportunity.

Learning Objectives (cont'd) Show how value additivity can be used to help managers maximize the value of the firm. Describe the Separation Principle. Calculate the value of a risky asset, using the Law of One Price. 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.

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

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.

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

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

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.

Example 3.1

Example 3.1 (cont'd)

Example 3.2

Example 3.2 (cont'd)

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.

Example 3.3

Example 3.3 (cont'd)

3.2 Interest Rates and 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 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)

Example 3.4

Example 3.4 (cont'd)

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: $100 million × (1.085) = $108.5 million Compare this amount to the cost of $100 million in 2006 using the interest rate of 4%: $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

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

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.

Example 3.5

Example 3.5 (cont'd)

Choosing Among Projects

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.

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.

NPV and Individual Preferences (cont'd)

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.

Figure 3.2 Comparing Projects A, B, & C

3.4 Arbitrage and the Law of One Price 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) 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 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

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.

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.

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

Example 3.6

Example 3.6 (cont'd)

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.

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%.

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

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.

Example 3.7

Example 3.7 (cont'd)

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

Example 3.8

Example 3.8 (cont'd)

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 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 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.

3.6 The Price of Risk (cont'd) Risky Versus Risk-free Cash Flows (cont’d) 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.

Risk Aversion and the Risk Premium 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.

Risk Aversion and the Risk Premium (cont’d) 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%

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.

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.

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

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.

Risk Is Relative to the Overall Market (cont'd)

Example 3.9

Example 3.9 (cont'd)

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.

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:

Example 3.10

Example 3.10 (cont'd)

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.

Example 3.11

Example 3.11 (cont'd)