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Berk Chapter 3: Arbitrage And Financial Decision Making
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Berk Chapter 3: Arbitrage And Financial Decision Making Berk Chapter 3: Arbitrage And Financial Decision Making Presentation Transcript

  • Chapter 3 Arbitrage and Financial Decision Making
  • 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
  • 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.
  • 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
  • Analyzing Costs and Benefits
    • 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.
  • Analyzing Costs and Benefits (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) X ($250/ounce) = $5000 today
  • Analyzing Costs and Benefits (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) X ($550/ounce) = $5500
  • Analyzing Costs and Benefits (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.
  • 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.
  • Textbook Example 3.2
  • Textbook Example 3.2 (cont'd)
  • Alternative Example 3.2
    • Problem
      • You are offered the following investment opportunity: In exchange for $27,000 today, you will receive 2,500 shares of stock in the Ford Motor Company and 10,000 euros today. The current market price for Ford stock is $9 per share and the current exchange rate is $1.50 per €. Should you take this opportunity? Would your decision change if you believed the value of the euro would rise over the next month?
  • Alternative Example 3.2 (cont'd)
    • Solution
      • The costs and benefits must be converted to their cash values. Assuming competitive market prices:
      • 2,500 shares × $9/share = $22,500
      • € 10,000 × $1.50/ € = $15,000
      • The net value of the opportunity is $22,500 + $15,000 - $40,000 = -$2,500, we should not take it. This value depends only on the current market prices for Ford and the euro. Our personal opinion about the future prospects of the euro and Ford does not alter the value the decision today.
  • 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), r f : The interest rate at which money can be borrowed or lent without risk.
          • Interest Rate Factor = 1 + r f
          • Discount Factor = 1 / (1 + r f )
  • 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
  • The Interest Rate: An Exchange Rate Across Time (cont'd)
    • 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.
  • The Interest Rate: An Exchange Rate Across Time (cont'd)
    • 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.
  • The Interest Rate: An Exchange Rate Across Time (cont'd)
    • 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.
  • The Interest Rate: An Exchange Rate Across Time (cont'd)
    • 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.
  • The Interest Rate: An Exchange Rate Across Time (cont'd)
    • Discount Factors and Rate
      • We can interpret
    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.
  • Textbook Example 3.3
  • Textbook Example 3.3 (cont'd)
  • Alternative Example 3.3
    • 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?
  • Alternative Example 3.3
    • 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.
  • 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.
  • Textbook Example 3.4
  • Textbook Example 3.4 (cont'd)
  • 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.
  • Textbook Example 3.5
  • Textbook Example 3.5 (cont'd)
  • 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
  • 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
  • Textbook Example 3.6
  • Textbook 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 and Firm Decision Making
    • In a normal market, the NPV of buying or selling a security is zero.
  • 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.
  • Textbook Example 3.7
  • Textbook 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
  • Textbook Example 3.8
  • Textbook 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?
  • 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
  • Chapter Quiz
    • If gasoline trades in a competitive market, would a transportation company that has a use for the gasoline value it differently than another investor?
    • How do you compare benefits at different points in time?
    • If interest rates fall, what happens to the value today of a promise of money in one year?
    • What is the NPV decision rule?
  • Chapter Quiz (cont'd)
    • Does the NPV decision rule depend on the investor’s preferences?
    • What is the Law of One Price?
    • What is Arbitrage?
    • If a firm makes an investment that has a negative NPV, how does the value of the firm change?
    • What is the Separation Principle?
  • Chapter 3 Appendix
  • Learning Objectives
    • 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.
  • 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
  • Appendix: 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
    • 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.
  • 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.
    Table 3A.2 Determining the Market Price of Security A (cash flows in $)
  • 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) Table 3A.3 Risk and Risk Premiums for Different Securities
  • Textbook Example 3.A.1
  • Textbook Example 3A.1 (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 r s that includes a risk premium appropriate for the investment’s risk:
  • 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:
      • r s = r f + (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
  • Textbook Example 3A.3 (cont'd)
  • Chapter Appendix Quiz
    • Why does the expected return of a risky security generally differ from the risk-free interest rate?
    • Should the risk of a security be evaluated in isolation?
    • In the presence of transactions costs, why might different investors disagree about the value of an investment opportunity?