Question 4 Time Value of Money PDF

Financial Decision CH APTE R Making and the Law of One Price 3 IN MID-2007, MICROSOFT DECIDED TO ENTER A BI...

Financial Decision CH APTE R Making and the Law of One Price 3 IN MID-2007, MICROSOFT DECIDED TO ENTER A BIDDING WAR with competitors Google and Yahoo! for a stake in the fast-growing social net- working site, Facebook. How did Microsoft’s managers decide that this was a NOTATION good decision? NPV net present value Every decision has future consequences that will affect the value of the firm. These consequences will generally include both benefits and costs. For example, r f   risk-free interest rate after raising its offer, Microsoft ultimately succeeded in buying a 1.6% stake in PV present value Facebook, along with the right to place banner ads on the Facebook Web site, for $240 million. In addition to this up-front payment, Microsoft also incurred ongo- ing expenses associated with software development for the advertising platform, network infrastructure, and international marketing efforts to attract advertisers. The benefits of the deal to Microsoft included the revenues associated with the advertising sales, together with the appreciation of its 1.6% stake in Facebook. In the end, Microsoft’s decision appeared to be a good one—in addition to advertis- ing benefits, by the time of Facebook’s IPO in May 2012, the value of Microsoft’s 1.6% stake had grown to over $1 billion. More generally, a decision is good for the firm’s investors if it increases the firm’s value by providing benefits whose total value exceeds their cost. But com- paring costs and benefits is often complicated because they occur at different points in time, may be in different currencies, or may have different risks associ- ated with them. To make a valid comparison, we must use the tools of finance to express all costs and benefits in common terms. In this chapter, we introduce a central principle of finance, which we name the Valuation Principle, which states that we can use current market prices to determine the value today of the costs and benefits associated with a decision. This principle allows us to apply the con- cept of net present value (NPV) as a way to compare the costs and benefits of a project in terms of a common unit—namely, dollars today. We will then be able to evaluate a decision by answering this question: Does the cash value today of its benefits exceed the cash value today of its costs? In addition, we will see that the NPV indicates the net amount by which the decision will increase wealth. 99 M03_BERK6318_06_GE_C03.indd 99 27/04/23 10:51 AM  100 Chapter 3 Financial Decision Making and the Law of One Price We then turn to financial markets and apply these same tools to determine the prices of securities that trade in the market. We discuss strategies called arbitrage, which allow us to exploit situations in which the prices of publicly available investment opportunities do not con- form to these values. Because investors trade rapidly to take advantage of arbitrage opportuni- ties, we argue that equivalent investment opportunities trading simultaneously in competitive markets must have the same price. This Law of One Price is the unifying theme of valuation that we use throughout this text. 3.1 Valuing Decisions A financial manager’s job is to make decisions on behalf of the firm’s investors. For ex- ample, when faced with an increase in demand for the firm’s products, a manager may need to decide whether to raise prices or increase production. If the decision is to raise production and a new facility is required, is it better to rent or purchase the facility? If the facility will be purchased, should the firm pay cash or borrow the funds needed to pay for it? In this book, our objective is to explain how to make decisions that increase the value of the firm to its investors. In principle, the idea is simple and intuitive: For good decisions, the benefits exceed the costs. Of course, real-world opportunities are usually complex and so the costs and benefits are often difficult to quantify. The analysis will often involve skills from other management disciplines, as in these examples: Marketing: to forecast the increase in revenues resulting from an advertising campaign Accounting: to estimate the tax savings from a restructuring Economics: to determine the increase in demand from lowering the price of a product Organizational Behavior : to estimate the productivity gains from a change in management structure Strategy: to predict a competitor’s response to a price increase Operations: to estimate the cost savings from a plant modernization Once the analysis of these other disciplines has been completed to quantify the costs and benefits associated with a decision, the financial manager must compare the costs and ben- efits and determine the best decision to make for the value of the firm. Analyzing Costs and Benefits The first step in decision making is to identify the costs and benefits of a decision. The next step is to quantify these costs and benefits. In order to compare the costs and benefits, we need to evaluate them in the same terms—cash today. Let’s make this concrete with a simple example. Suppose a jewelry manufacturer has the opportunity to trade 400 ounces of silver for 10 ounces of gold today. Because an ounce of gold differs in value from an ounce of silver, it is incorrect to compare 400 ounces to 10 ounces and conclude that the larger quantity is better. Instead, to compare the costs and benefits, we first need to quantify their values in equivalent terms. M03_BERK6318_06_GE_C03.indd 100 27/04/23 10:51 AM  3.1 Valuing Decisions 101 Consider the silver. What is its cash value today? Suppose silver can be bought and sold for a current market price of $25 per ounce. Then the 400 ounces of silver we give up has a cash value of   1 ( 400 ounces of silver today ) × ( $25 ounce of silver today ) = $10,000 today If the current market price for gold is $1900 per ounce, then the 10 ounces of gold we receive has a cash value of ( 10 ounces of gold today ) × ( $1900 ounce of gold today ) = $19,000 today Now that we have quantified the costs and benefits in terms of a common measure of value, cash today, we can compare them. The jeweler’s opportunity has a benefit of $19,000 today and a cost of $10,000 today, so the net value of the decision is $19,000 − $10,000 = $9000 today. By accepting the trade, the jewelry firm will be richer by $9000. Using Market Prices to Determine Cash Values In evaluating the jeweler’s decision, we used the current market price to convert from ounces of silver 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. Do such considerations matter? Suppose, for example, that the jeweler does not need the gold, or thinks the current price of gold is too high. Would he value the gold at less than $19,000? The answer is no—he can always sell the gold at the current market price and receive $19,000 right now. Similarly, he would not value the gold at more than $19,000, because even if he really needs the gold or thinks the current price of gold is too low, he can always buy 10 ounces of gold for $19,000. Thus, independent of his own views or preferences, the value of the gold to the jeweler is $19,000. This example illustrates an important general principle: Whenever a good trades in a competitive market—by which we mean a market in which it can be bought and sold at the same price—that price determines the cash value of the good. As long as a competitive market exists, the value of the good will not depend on the views or preferences of the decision maker. EXAMPLE 3.1 Competitive Market Prices Determine Value Problem You have just won a radio contest and are disappointed to find out that the prize is four tickets to the Def Leppard reunion tour (face value $40 each). Not being a fan of 1980s power rock, you have no intention of going to the show. However, there is a second choice: two tickets to your favorite band’s sold-out show (face value $45 each). You notice that tickets to the Def Leppard show are being bought and sold online for $30 apiece and tickets to your favorite band’s show are being bought and sold at $50 each. Which prize should you choose? 1 You might worry about commissions or other transactions costs that are incurred when buying or selling gold, in addition to the market price. For now, we will ignore transactions costs, and discuss their effect in the appendix to this chapter. M03_BERK6318_06_GE_C03.indd 101 27/04/23 10:51 AM  102 Chapter 3 Financial Decision Making and the Law of One Price Solution Competitive market prices, not your personal preferences (nor the face value of the tickets), are relevant here: Four Def Leppard tickets at $30 apiece = $120 market value Two of your favorite band’s tickets at $50 apiece = $100 market value Instead of taking the tickets to your favorite band, you should accept the Def Leppard tickets, sell them, and use the proceeds to buy two tickets to your favorite band’s show. You’ll even have $20 left over to buy a T-shirt. Thus, by evaluating cost and benefits using competitive market prices, we can determine whether a decision will make the firm and its investors wealthier. This point is one of the central and most powerful ideas in finance, which we call the Valuation Principle: The value of an asset to the firm or its investors is determined by its competitive market price. The benefits and costs of a decision should be evaluated using these market prices, and when the value of the benefits exceeds the value of the costs, the decision will increase the market value of the firm. The Valuation Principle provides the basis for decision making throughout this text. In the remainder of this chapter, we first apply it to decisions whose costs and benefits occur at different points in time and develop the main tool of project evaluation, the Net Present Value Rule. We then consider its consequences for the prices of assets in the market and develop the concept of the Law of One Price. EXAMPLE 3.2 Applying the Valuation Principle Problem You are the operations manager at your firm. Due to a pre-existing contract, you have the oppor- tunity to acquire 125 barrels of oil and 1500 pounds of copper for a total of $12,000. The cur- rent competitive market price of oil is $80 per barrel and for copper is $4 per pound. You are not sure you need all of the oil and copper, and are concerned that the value of both commodities may fall in the future. Should you take this opportunity? Solution To answer this question, you need to convert the costs and benefits to their cash values using market prices: ( 125 barrels of oil ) × ( $80 barrel of oil today ) = $10,000 today ( 1500 pounds of copper ) × ( $4 pound of copper today ) = $6000 today The net value of the opportunity is $10,000 + $6000 − $12,000 = $4000 today. Because the net value is positive, you should take it. This value depends only on the current market prices for oil and copper. Even if you do not need all the oil or copper, or expect their values to fall, you can sell them at current market prices and obtain their value of $16,000. Thus, the opportunity is a good one for the firm, and will increase its value by $4000. M03_BERK6318_06_GE_C03.indd 102 27/04/23 10:51 AM  3.2 Interest Rates and the Time Value of Money 103 When Competitive Market Prices Are Not Available Competitive market prices allow us to calculate the value for individuals opening a new account. At the time, the retail of a decision without worrying about the tastes or opinions price of that model iPad was $329. But because there is no of the decision maker. When competitive prices are not competitive market to trade iPads, the value of the iPad available, we can no longer do this. Prices at retail stores, depends on whether you were going to buy one or not. for example, are one sided: You can buy at the posted price, If you planned to buy the iPad anyway, then the value but you cannot sell the good to the store at that same price. to you is $329, the price you would otherwise pay for it. We cannot use these one-sided prices to determine an exact But if you did not want or need the iPad, the value of the cash value. They determine the maximum value of the good offer would depend on the price you could get for the (since it can always be purchased at that price), but an indi- iPad. For example, if you could sell the iPad for $250 to vidual may value it for much less depending on his or her your friend, then RBC’s offer is worth at least $250 to you. preferences for the good. Thus, depending on your preferences, RBC’s offer is worth Let’s consider an example. It has long been common somewhere between $250 (you don’t want an iPad) and for banks to entice new depositors by offering free gifts for $329 (you definitely want one). opening a new account. In 2021, RBC offered a free iPad CONCEPT CHECK 1. In order to compare the costs and benefits of a decision, what must we determine? 2. If crude oil trades in a competitive market, would an oil refiner that has a use for the oil value it differently than another investor? 3.2 Interest Rates and the Time Value of Money For most financial decisions, unlike in the examples presented so far, costs and benefits occur at different points in time. For example, typical investment projects incur costs up front and provide benefits in the future. In this section, we show how to account for this time difference when evaluating a project. 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 Because both are expressed in dollar terms, it might appear that the cost and benefit are directly comparable so that the project’s net value is $105,000 − $100,000 = $5000. But this calculation ignores the timing of the costs and benefits, and it treats money today as equivalent to money in one year. In general, a dollar today is worth more than a dollar in one year. If you have $1 today, you can invest it. For example, if you deposit it in a bank account paying 7% interest, you will have $1.07 at the end of one year. We call the difference in value between money today and money in the future the time value of money. The Interest Rate: An Exchange Rate Across Time By depositing money into a savings account, we can convert money today into money in the future with no risk. Similarly, by borrowing money from the bank, we can exchange money in the future for money today. The rate at which we can exchange money today for money in the future is determined by the current interest rate. In the same way that M03_BERK6318_06_GE_C03.indd 103 27/04/23 10:51 AM  104 Chapter 3 Financial Decision Making and the Law of One Price an exchange rate allows us to convert money from one currency to another, the interest rate allows us to convert money from one point in time to another. In essence, an interest rate is like an exchange rate across time. It tells us the market price today of money in the future. 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 $1 today. More generally, we define the risk-free interest rate, r f , for a given period as the interest rate at which money can be borrowed or lent without risk over that period. We can exchange (1 + r f ) dollars in the future per dollar today, and vice versa, without risk. We refer to (1 + r f ) as the interest rate factor for risk- free cash flows; it defines the exchange rate across time, and has units of “$ in one year/$ today.” As with other market prices, the risk-free interest rate depends on supply and demand. In particular, at the risk-free interest rate the supply of savings equals the demand for bor- rowing. After we know the risk-free interest rate, we can use it to evaluate other decisions in which costs and benefits are separated in time without knowing the investor’s preferences. Value of Investment in One Year. Let’s reevaluate the investment we considered earlier, this time taking into account the time value of money. 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 Think of this amount as the opportunity cost of spending $100,000 today: We give up the $107,000 we would have had in one year if we had left the money in the bank. Alternatively, if we were to borrow the $100,000, we would owe $107,000 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: If we took it, we would be $2000 poorer in one year than if we didn’t. Value of Investment Today. The previous calculation expressed the value of the costs and benefits in terms of dollars in one year. Alternatively, we can use the interest rate fac- tor to convert to dollars today. Consider the benefit of $105,000 in one year. What is the equivalent amount in terms of dollars today? That is, how much would we need to have in the bank today so that we would end up with $105,000 in the bank in one year? We find this amount by dividing by the interest rate factor: Benefit = ( $105,000 in one year ) ÷ ( 1.07 $ in one year $ today ) 1 = $105,000 × today 1.07 = $98,130.84 today This is also the amount the bank would lend to us today if we promised to repay $105,000 in one year.2 Thus, it is the competitive market price at which we can “buy” or “sell” $105,000 in one year. 2 We are assuming the bank will both borrow and lend at the risk-free interest rate. We discuss the case when these rates differ in “Arbitrage with Transactions Costs” in the appendix to this chapter. M03_BERK6318_06_GE_C03.indd 104 27/04/23 10:51 AM  3.2 Interest Rates and the Time Value of Money 105 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. Taking the investment would make us $1869.16 poorer today because we have given up $100,000 for something worth only $98,130.84. Present Versus Future Value. This calculation 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: We should reject the investment. Indeed, if we convert from dollars today to dollars in one year, ( −$1869.16 today ) × ( 1.07 $ in one year $ today ) = −$2000 in one year we see that the two results are equivalent, but expressed as values at different points in time. 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 Rates. When computing a present value as in the preceding cal- culation, we can interpret the term 1 1 = = 0.93458 $ today $ in one year 1+ r 1.07 as the price today of $1 in one year. Note that the value is less than $1—money in the future is worth less today, and so its price reflects a discount. Because it provides the discount at which we can purchase money in the future, the amount 1 +1 r is called the one-year discount factor. The risk-free interest rate is also referred to as the discount rate for a risk-free investment. EXAMPLE 3.3 Comparing Costs at Different Points in Time Problem The cost of rebuilding the San Francisco Bay Bridge to make it earthquake-safe was approxi- mately $3 billion in 2004. At the time, engineers estimated that if the project were delayed to 2005, the cost would rise by 10%. If the interest rate were 2%, what would be the cost of a delay in terms of dollars in 2004? Solution If the project were delayed, it would cost $3 billion × 1.10 = $3.3 billion in 2005. To compare this amount to the cost of $3 billion in 2004, we must convert it using the interest rate of 2%: $3.3 billion in 2005 ÷ ( $1.02 in 2005 $ in 2004 ) = $3.235 billion in 2004 Therefore, the cost of a delay of one year was $3.235 billion − $3 billion = $235 million in 2004 That is, delaying the project for one year was equivalent to giving up $235 million in cash. M03_BERK6318_06_GE_C03.indd 105 27/04/23 10:51 AM  106 Chapter 3 Financial Decision Making and the Law of One Price FIGURE 3.1 4 Gold Price (+/oz) Ounces of Converting between Gold Today Dollars Today and 3 Gold Price (+/oz) Gold, Euros, or Dollars in the Future 3 Exchange Rate (:/+) Dollars Today Euros Today We can convert dollars today to different goods, 4 Exchange Rate (:/+) currencies, or points in time 3 (1 1 r f ) by using the competitive market price, exchange Dollars in One Year rate, or interest rate. 4 (1 1 r f ) We can use the risk-free interest rate to determine values in the same way we used competitive market prices. Figure 3.1 illustrates how we use competitive market prices, exchange rates, and interest rates to convert between dollars today and other goods, cur- rencies, or dollars in the future. CONCEPT CHECK 1. How do you compare costs at different points in time? 2. If interest rates rise, what happens to the value today of a promise of money in one year? 3.3 Present Value and the NPV Decision Rule In Section 3.2, we converted between cash today and cash in the future using the risk-free interest rate. As long as we convert costs and benefits to the same point in time, we can compare them to make a decision. In practice, however, most corporations prefer to mea- sure values in terms of their present value—that is, in terms of cash today. In this section we apply the Valuation Principle to derive the concept of the net present value, or NPV, and define the “golden rule” of financial decision making, the NPV Rule. Net Present Value When we compute the value of a cost or benefit in terms of cash today, we refer to it as the present value (PV). Similarly, we define the net present value (NPV) of a project or investment as the difference between the present value of its benefits and the present value of its costs: Net Present Value NPV = PV (Benefits) − PV (Costs) (3.1) If we use positive cash flows to represent benefits and negative cash flows to represent costs, and calculate the present value of multiple cash flows as the sum of present values for individual cash flows, we can also write this definition as NPV = PV ( All project cash flows) (3.2) That is, the NPV is the total of the present values of all project cash flows. M03_BERK6318_06_GE_C03.indd 106 27/04/23 10:51 AM  3.5 No-Arbitrage and Security Prices 111 of it. Those investors who spot the opportunity first and who can trade quickly will have the ability to exploit it. Once they place their trades, prices will respond, causing the arbi- trage opportunity to evaporate. Arbitrage opportunities are like money lying in the street; once spotted, they will quickly disappear. Thus the normal state of affairs in markets should be that no arbitrage oppor- tunities exist. We call a competitive market in which there are no arbitrage opportunities a normal market.6 Law of One Price In a normal market, the price of gold at any point in time will be the same in London and New York. The same logic applies more generally whenever equivalent investment oppor- tunities trade in two different competitive markets. If the prices in the two markets differ, investors will profit immediately by buying in the market where it is cheap and selling in the market where it is expensive. In doing so, they will equalize the prices. As a result, prices will not differ (at least not for long). This important property is the Law of One Price: If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in all markets. One useful consequence of the Law of One Price is that when evaluating costs and benefits to compute a net present value, we can use any competitive price to determine a cash value, without checking the price in all possible markets. CONCEPT CHECK 1. If the Law of One Price were violated, how could investors profit? 2. When investors exploit an arbitrage opportunity, how do their actions affect prices? 3.5 No-Arbitrage and Security Prices An investment opportunity that trades in a financial market is known as a financial ­security (or, more simply, a security). The notions of arbitrage and the Law of One Price have important implications for security prices. We begin exploring its implications for the prices of individual securities as well as market interest rates. We then broaden our per- spective to value a package of securities. Along the way, we will develop some important insights about firm decision making and firm value that will underpin our study throughout this textbook. Valuing a Security with the Law of One Price The Law of One Price tells us that the prices of equivalent investment opportunities should be the same. We can use this idea to value a security if we can find another equivalent in- vestment whose price is already known. Consider a simple security that promises a one- time payment to its owner of $1000 in one year’s time. Suppose there is no risk that the 6 The term efficient market is also sometimes used to describe a market that, along with other properties, is without arbitrage opportunities. We avoid that term here because it is stronger than we require, as it also restricts the information held by market participants. We discuss notions of market efficiency in Chapter 9. M03_BERK6318_06_GE_C03.indd 111 27/04/23 10:51 AM  112 Chapter 3 Financial Decision Making and the Law of One Price payment will not be made. One example of this type of security is a bond, a security sold by governments and corporations to raise money from investors today in exchange for the promised future payment. If the risk-free interest rate is 5%, what can we conclude about the price of this bond in a normal market? To answer this question, consider an alternative investment that would generate the same cash flow as this bond. Suppose we invest money at the bank at the risk-free ­interest rate. How much do we need to invest today to receive $1000 in one year? As we saw in Section 3.3, the cost today of recreating a future cash flow on our own is its present value: PV ($1000 in one year ) = ( $1000 in one year ) ÷ ( 1.05 $ in one year $ today ) = $952.38 today If we invest $952.38 today at the 5% risk-free interest rate, we will have $1000 in one year’s time with no risk. We now have two ways to receive the same cash flow: (1) buy the bond or (2) invest $952.38 at the 5% risk-free interest rate. Because these transactions produce equivalent cash flows, the Law of One Price implies that, in a normal market, they must have the same price (or cost). Therefore, Price(Bond) = $952.38 Identifying Arbitrage Opportunities with Securities. Recall that the Law of One Price is based on the possibility of arbitrage: If the bond had a different price, there would be an arbitrage opportunity. For example, suppose the bond traded for a price of $940. How could we profit in this situation? In this case, we can buy the bond for $940 and at the same time borrow $952.38 from the bank. Given the 5% interest rate, we will owe the bank $952.38 × 1.05 = $1000 in one year. Our overall cash flows from this pair of transactions are as shown in Table 3.3. Using this strategy we can earn $12.38 in cash today for each bond that we buy, without taking any risk or paying any of our own money in the future. Of course, as we—and others who see the opportunity—start buying the bond, its price will quickly rise until it reaches $952.38 and the arbitrage opportunity disappears. A similar arbitrage opportunity arises if the bond price is higher than $952.38. For e­ xample, suppose the bond is trading for $960. In that case, we should sell the bond and invest $952.38 at the bank. As shown in Table 3.4, we then earn $7.62 in cash today, yet keep our future cash flows unchanged by replacing the $1000 we would have received from the bond with the $1000 we will receive from the bank. Once again, as people begin selling the bond to exploit this op- portunity, the price will fall until it reaches $952.38 and the arbitrage opportunity disappears. TABLE 3.3  Net Cash Flows from Buying the Bond and Borrowing Today ($) In One Year ($) Buy the bond −940.00 +1000.00 Borrow from the bank +952.38 −1000.00 Net cash flow +12.38 0.00 M03_BERK6318_06_GE_C03.indd 112 27/04/23 10:51 AM  3.5 No-Arbitrage and Security Prices 113 TABLE 3.4  et Cash Flows from Selling the Bond N and Investing Today ($) In One Year ($) Sell the bond +960.00 −1000.00 Invest at the bank −952.38 +1000.00 Net cash flow +7.62 0.00 When the bond is overpriced, the arbitrage strategy involves selling the bond and invest- ing some of the proceeds. But if the strategy involves selling the bond, does this mean that only the current owners of the bond can exploit it? The answer is no; in financial markets it is possible to sell a security you do not own by doing a short sale. In a short sale, the person who intends to sell the security first borrows it from someone who already owns it. Later, that person must either return the security by buying it back or pay the owner the cash flows he or she would have received. For example, we could short sell the bond in the example ef- fectively promising to repay the current owner $1000 in one year. By executing a short sale, it is possible to exploit the arbitrage opportunity when the bond is overpriced even if you do not own it. Determining the No-Arbitrage Price. We have shown that at any price other than $952.38, an arbitrage opportunity exists for our bond. Thus, in a normal market, the price of this bond must be $952.38. We call this price the no-arbitrage price for the bond. By applying the reasoning for pricing the simple bond, we can outline a general process for pricing other securities: 1. Identify the cash flows that will be paid by the security. 2. Determine the “do-it-yourself ” cost of replicating those cash flows on our own; that is, the present value of the security’s cash flows. Unless the price of the security equals this present value, there is an arbitrage opportunity. Thus, the general formula is No-Arbitrage Price of a Security Price(Security) = PV ( All cash flows paid by the security ) (3.3) Determining the Interest Rate from Bond Prices. Given the risk-free interest rate, the no-arbitrage price of a risk-free bond is determined by Eq. 3.3. The reverse is also true: If we know the price of a risk-free bond, we can use Eq. 3.3 to determine what the risk-free interest rate must be if there are no arbitrage opportunities. For example, suppose a risk-free bond that pays $1000 in one year is currently trading with a competitive market price of $929.80 today. From Eq. 3.3, we know that the bond’s price equals the present value of the $1000 cash flow it will pay: $929.80 today = ( $1000 in one year ) ÷ ( 1 + r f ) M03_BERK6318_06_GE_C03.indd 113 27/04/23 10:51 AM  114 Chapter 3 Financial Decision Making and the Law of One Price We can rearrange this equation to determine the risk-free interest rate: $1000 in one year 1+ rf = = 1.0755 $ in one year $ today $929.80 today That is, if there are no arbitrage opportunities, the risk-free interest rate must be 7.55%. Interest rates are calculated by this method in practice. Financial news services report current interest rates by deriving these rates based on the current prices of risk-free gov- ernment bonds trading in the market. Note that the risk-free interest rate equals the percentage gain that you earn from invest- ing in the bond, which is called the bond’s return: Gain at End of Year Return = Initial Cost 1000 − 929.80 1000 = = − 1 = 7.55% (3.4) 929.80 929.80 Thus, if there is no arbitrage, the risk-free interest rate is equal to the return from ­investing in a risk-free bond. If the bond offered a higher return than the ­risk-free ­interest rate, then investors would earn a profit by borrowing at the risk-free interest rate and investing in the bond. If the bond had a lower return than the risk-free i­nterest rate, investors would sell the bond and invest the proceeds at the risk-free interest rate. No arbitrage is therefore equivalent to the idea that all risk-free investments should offer investors the same return. EXAMPLE 3.6 Computing the No-Arbitrage Price or Interest Rate Problem Consider a security that pays its owner $100 today and $100 in one year, without any risk. Suppose the risk-free interest rate is 10%. What is the no-arbitrage price of the security today (before the first $100 is paid)? If the security is trading for $195, what arbitrage opportunity is available? At what interest rate would the arbitrage opportunity disappear? Solution We need to compute the present value of the security’s cash flows. In this case there are two cash flows: $100 today, which is already in present value terms, and $100 in one year. The present value of the second cash flow is $100 in one year ÷ 1.10 $ in one year $ today = $90.91 today Therefore, the total present value of the cash flows is $100 + $90.91 = $190.91 today, which is the no-arbitrage price of the security. If the security is trading for $195, we can exploit its overpricing by selling it for $195. We can then use $100 of the sale proceeds to replace the $100 we would have received from the security today and invest $90.91 of the sale proceeds at 10% to replace the $100 we would have received in one year. The remaining $195 − $100 − $90.91 = $4.09 is an arbitrage profit. At a price of $195, we are effectively paying $95 to receive $100 in one year. So, an arbitrage opportunity exists unless the interest rate equals 100/95 − 1 = 5.26%. M03_BERK6318_06_GE_C03.indd 114 27/04/23 10:51 AM  3.5 No-Arbitrage and Security Prices 115 An Old Joke There is an old joke that many finance professors enjoy then asks whether anyone has ever actually found a real $100 telling their students. It goes like this: bill lying on the pavement. The ensuing silence is the real lesson behind the joke. A finance professor and a student are walking down a street. This joke sums up the point of focusing on markets in The student notices a $100 bill lying on the pavement and leans which no arbitrage opportunities exist. Free $100 bills lying down to pick it up. The finance professor immediately intervenes on the pavement, like arbitrage opportunities, are extremely and says, “Don’t bother; there is no free lunch. If there was a rare for two reasons: (1) Because $100 is a large amount of real $100 bill lying there, somebody would already have picked money, people are especially careful not to lose it, and (2) in it up!” the rare event when someone does inadvertently drop $100, This joke invariably generates much laughter because it the likelihood of your finding it before someone else does is makes fun of the principle of no arbitrage in competitive extremely small. markets. But once the laughter dies down, the professor The NPV of Trading Securities and Firm Decision Making We have established that positive-NPV decisions increase the wealth of the firm and its investors. Think of buying a security as an investment decision. The cost of the decision is the price we pay for the security, and the benefit is the cash flows that we will receive from owning the security. When securities trade at no-arbitrage prices, what can we conclude about the value of trading them? From Eq. 3.3, the cost and benefit are equal in a normal market and so the NPV of buying a security is zero: NPV (Buy security ) = PV ( All cash flows paid by the security ) − Price(Security) =0 Similarly, if we sell a security, the price we receive is the benefit and the cost is the cash flows we give up. Again the NPV is zero: NPV (Sell security ) = Price(Security) − PV ( All cash flows paid by the security ) =0 Thus, the NPV of trading a security in a normal market is zero. This result is not sur- prising. If the NPV of buying a security were positive, then buying the security would be equivalent to receiving cash today—that is, it would present an arbitrage opportunity. Because arbitrage opportunities do not exist in normal markets, the NPV of all security trades must be zero. Another way to understand this result is to remember that every trade has both a buyer and a seller. In a competitive market, if a trade offers a positive NPV to one party, it must give a negative NPV to the other party. But then one of the two parties would not agree to the trade. Because all trades are voluntary, they must occur at prices at which neither party is losing value, and therefore for which the trade is zero NPV. The insight that security trading in a normal market is a zero-NPV transaction is a criti- cal building block in our study of corporate finance. Trading securities in a normal market neither creates nor destroys value: Instead, value is created by the real investment projects in which the firm engages, such as developing new products, opening new stores, or creat- ing more efficient production methods. Financial transactions are not sources of value but instead serve to adjust the timing and risk of the cash flows to best suit the needs of the firm or its investors. M03_BERK6318_06_GE_C03.indd 115 27/04/23 10:51 AM  116 Chapter 3 Financial Decision Making and the Law of One Price An important consequence of this result is the idea that we can evaluate a decision by focusing on its real components, rather than its financial ones. That is, we can separate the firm’s investment decision from its financing choice. We refer to this concept as the Separation Principle: Security transactions in a normal market neither create nor destroy value on their own. Therefore, 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 Separating Investment and Financing Problem Your firm is considering a project that will require an up-front investment of $10 million today and will produce $12 million in cash flow for the firm in one year without risk. Rather than pay for the $10 million investment entirely using its own cash, the firm is considering raising addi- tional funds by issuing a security that will pay investors $5.5 million in one year. Suppose the risk-free interest rate is 10%. Is pursuing this project a good decision without issuing the new security? Is it a good decision with the new security? Solution Without the new security, the cost of the project is $10 million today and the benefit is $12 mil- lion in one year. Converting the benefit to a present value $12 million in one year ÷ ( 1.10 $ in one year $ today ) = $10.91 million today we see that the project has an NPV of $10.91 million − $10 million = $0.91 million today. Now suppose the firm issues the new security. In a normal market, the price of this security will be the present value of its future cash flow: Price(Security) = $5.5 million ÷ 1.10 = $5 million today Thus, after it raises $5 million by issuing the new security, the firm will only need to invest an ad- ditional $5 million to take the project. To compute the project’s NPV in this case, note that in one year the firm will receive the $12 million payout of the project, but owe $5.5 million to the investors in the new security, leaving $6.5 million for the firm. This amount has a present value of $6.5 million in one year ÷ ( 1.10 $ in one year $ today ) = $5.91 million today Thus, the project has an NPV of $5.91 million − $5 million = $0.91 million today, as before. In either case, we get the same result for the NPV. The separation principle indicates that we will get the same result for any choice of financing for the firm that occurs in a normal market. We can therefore evaluate the project without explicitly considering the different financing possibili- ties the firm might choose. Valuing a Portfolio So far, we have discussed the no-arbitrage price for individual securities. 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, se- curity C is equivalent to a combination of the securities A and B. We use the term portfolio to describe a collection of securities. What can we conclude about the price of security C as compared to the prices of A and B? M03_BERK6318_06_GE_C03.indd 116 27/04/23 10:51 AM  3.5 No-Arbitrage and Security Prices 117 Value Additivity. Because security C is equivalent to the portfolio of A and B, by the Law of One Price, they must have the same price. This idea leads to the relationship known as value additivity; that is, the price of C must equal the price of the portfolio, which is the combined price of A and B: Value Additivity Price(C) = Price(A + B) = Price(A ) + Price(B) (3.5) Because security C has cash flows equal to the sum of A and B, its value or price must be the sum of the values of A and B. Otherwise, an obvious arbitrage opportunity would exist. For example, if the total price of A and B were lower than the price of C, then we could make a profit buying A and B and selling C. This arbitrage activity would quickly push prices until the price of security C equals the total price of A and B. EXAMPLE 3.8 Valuing an Asset in a Portfolio Problem Holbrook Holdings is a publicly traded company with only two assets: It owns 60% of Harry’s Hotcakes restaurant chain and an ice hockey team. Suppose the market value of Holbrook Hold- ings is $160 million, and the market value of the entire Harry’s Hotcakes chain (which is also publicly traded) is $120 million. What is the market value of the hockey team? Solution We can think of Holbrook as a portfolio consisting of a 60% stake in Harry’s Hotcakes and the hockey team. By value additivity, the sum of the value of the stake in Harry’s Hotcakes and the hockey team must equal the $160 million market value of Holbrook. Because the 60% stake in Harry’s Hotcakes is worth 60% × $120 million = $72 million, the hockey team has a value of $160 million − $72 million = $88 million. FINANCE IN TIMES OF DISRUPTION Liquidity and the Informational Role of Prices In 2007, a decade-long boom in U.S. house prices abruptly it became impossible to reliably value these securities. In addi- ended and prices steeply declined. As the severity of the tion, given that the value of the banks holding these securities downturn became apparent, investors grew increasingly was based on the sum of all projects and investments within concerned about the risk that homeowners might default them, investors could not value the banks either. Investors re- on their mortgages. As a result, the volume of trade in acted to this uncertainty by selling both the mortgage-backed the multi-trillion dollar market for mortgage-backed se- securities and securities of banks that held mortgage-backed curities plummeted over 80% by August 2008. Over the securities. These actions further compounded the problem by next two months, trading in many of these securities driving down prices to seemingly unrealistically low levels and ceased altogether, making the markets for these securities thereby threatening the solvency of the entire financial system. increasingly illiquid. The loss of information precipitated by the loss of li- Competitive markets depend upon liquidity—there must quidity played a key role in the breakdown of credit mar- be sufficient buyers and sellers of a security so that it is pos- kets. As both investors and government regulators found sible to trade at any time at the current market price. When it increasingly difficult to assess the solvency of the banks, markets become illiquid it may not be possible to trade at the banks found it difficult to raise new funds on their own and posted price. As a consequence, we can no longer rely on also shied away from lending to other banks because of market prices as a measure of value. their concerns about the financial viability of their competi- The collapse of the mortgage-backed securities market cre- tors. The result was a breakdown in lending. Ultimately, the ated two problems. First was the loss of trading opportunities, government was forced to step in and spend hundreds of making it difficult for holders of these securities to sell them. billions of dollars in order to (1) provide new capital to sup- But a potentially more significant problem was the loss of infor- port the banks and (2) provide liquidity by creating a market mation. Without a liquid, competitive market for these securities, for the now “toxic” mortgage-backed securities. M03_BERK6318_06_GE_C03.indd 117 27/04/23 10:51 AM  118 Chapter 3 Financial Decision Making and the Law of One Price Arbitrage in Markets Value additively is the principle behind a type of trading activ- The evolution of how traders took advantage of these ity known as stock index arbitrage. Common stock indices short-lived arbitrage opportunities provides a nice illustration (such as the Dow Jones Industrial Average and the Standard of how competitive market forces act to remove profit-­making and Poor’s 500 (S&P 500)) represent portfolios of individual opportunities. In a recent study, Professors Eric Budish, Peter stocks. It is possible to trade the individual stocks that com- Crampton, and John Shim† focused on the evolution of one prise an index on the New York Stock Exchange and Nasdaq. particular arbitrage opportunity that resulted from differences It is also possible to trade the entire index (as a single security) in the price of the S&P 500 Futures Contract on the Chicago on the futures exchanges in Chicago, or as an exchange-traded Mercantile Exchange and the price of the SPDR S&P 500 fund (ETF) on the NYSE. When the price of the index secu- ETF traded on the New York Stock Exchange. rity is below the total price of the individual stocks, traders The left figure shows how the duration of arbitrage buy the index and sell the stocks to capture the price differ- opportunities changed between 2005 and 2011. Each line ence. Similarly, when the price of the index security is above shows, for the indicated year, the fraction of arbitrage the total price of the individual stocks, traders sell the index opportunities that lasted longer than the amount of time and buy the individual stocks. In 2012 it was not uncommon indicated on the horizontal axis. So, for example, in 2005 for almost half of the daily volume of trade on the NYSE to about half of the arbitrage opportunities that existed be due to index arbitrage activity via program trading.* lasted more than 100 milliseconds. By 2008, this number The traders that engage in stock index arbitrage automate the had dropped to 20 milliseconds, and by 2011, the number process by tracking prices and submitting (or cancelling) orders was under 10 milliseconds. Note also that in 2005 almost electronically. Over the years the competition to take advantage all opportunities lasted at least 20 milliseconds, but by of these opportunities has caused traders to go to extraordinary 2011 the number of opportunities that lasted this long was lengths to reduce order execution time. One limiting factor is less than 10% and hardly any persisted for more than 100 the time it takes to send an order from one exchange to another. milliseconds. For example, in 2010 Spread Networks paid $300 million for What happened to the profits from exploiting these mis- a new fiber optic line that reduced the communication time pricings? You might have expected that the effect of this com- between New York and Chicago exchanges from 16 millisec- petition would be to decrease profits, but as the right figure onds to 13 milliseconds. Three milliseconds might not sound shows, profits per opportunity remained relatively constant. like a lot (it takes 400 milliseconds to blink), but it meant that Furthermore, the number of opportunities did not systemati- Spread would be able to exploit any mispricings between the cally decline over this period, implying that the aggregate profits exchanges before its competitors, at least until one of them from exploiting arbitrage opportunities did not diminish. In that constructed a faster line. (Indeed, the Spread advantage did sense, the competition between arbitrageurs has not reduced not last long. Within a year, traders began to use microwaves the magnitude or frequency of price deviations across these to transmit information in a straight line through the earth, ulti- markets, but instead has reduced the amount of time that these mately cutting the communication time to 8 milliseconds.). deviations can persist. Duration of Arbitrage Opportunities Median Profit per Arbitrage Opportunity Proportion of Arbitrage Opportunities 1 0.15 Lasting Longer Than This Duration 2005 2009 0.9 (index points per unit traded) 2006 2010 Average Profit per Arbitrage 2007 2011 0.125 0.8 2008 0.7 0.1 0.6 0.5 0.075 0.4 0.05 0.3 0.2 0.025 0.1 0 0 20 40 60 80 100 2005 2006 2007 2008 2009 2010 2011 2012 Duration of Arbitrage Opportunity (ms) Date *See https://www.nyse.com/publicdocs/nyse/markets/nyse/PT122812.pdf †“The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response,” Quarterly Journal of Economics (2015): 1547–1621. M03_BERK6318_06_GE_C03.indd 118 27/04/23 10:51 AM  3.5 No-Arbitrage and Security Prices 119 More generally, value additivity implies that the value of a portfolio is equal to the sum of the values of its parts. That is, the “à la carte” price and the package price must coincide.7 Value Additivity and Firm Value. Value additivity has an important consequence for the value of an entire firm. The cash flows of the firm are equal to the total cash flows of all projects and investments within the firm. Therefore, by value additivity, the price or value of the entire firm is equal to the sum of the values of all projects and investments within it. In other words, our NPV decision rule coincides with maximizing the value of the entire firm: 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. Where Do We Go from Here? The key concepts we have developed in this chapter—the Valuation Principle, Net Present Value, and the Law of One Price—provide the foundation for financial decision making. The Law of One Price allows us to determine the value of stocks, bonds, and other secu- rities, based on their cash flows, and validates the optimality of the NPV decision rule in identifying projects and investments that create value. In the remainder of the text, we will build on this foundation and explore the details of applying these principles in practice. For simplicity, we have focused in this chapter on projects that were not risky, and thus had known costs and benefits. The same fundamental tools of the Valuation Principle and the Law of One Price can be applied to analyze risky investments as well, and we will look in detail at methods to assess and value risk in Part 4 of the text. Those seeking some early insights and key foundations for this topic, however, are strongly encouraged to read the appendix to this chapter. There we introduce the idea that investors are risk averse, and then use the principle of no-arbitrage developed in this chapter to demonstrate two funda- mental insights regarding the impact of risk on valuation: 1. When cash flows are risky, we must discount them at a rate equal to the risk-free interest rate plus an appropriate risk premium; and, 2. The appropriate risk premium will be higher the more the project’s returns tend to vary with the overall risk in the economy. Finally, the chapter appendix also addresses the important practical issue of ­transactions costs. There we show that when purchase and sale prices, or borrowing and lend- ing rates differ, the Law of One Price will continue to hold, but only up to the level of transactions costs. CONCEPT CHECK 1. If a firm makes an investment that has a positive NPV, how does the value of the firm change? 2. What is the Separation Principle? 3. In addition to trading opportunities, what else do liquid markets provide? 7 This feature of financial markets does not hold in many other noncompetitive markets. For example, a round-trip airline ticket often costs much less than two separate one-way tickets. Of course, airline tickets are not sold in a competitive market—you cannot buy and sell the tickets at the listed prices. Only airlines can sell tickets, and they have strict rules against reselling tickets. Otherwise, you could make money buy- ing round-trip tickets and selling them to people who need one-way tickets. M03_BERK6318_06_GE_C03.indd 119 27/04/23 10:51 AM  120 Chapter 3 Financial Decision Making and the Law of One Price Key Points 3.1 Valuing Decisions and Equations To evaluate a decision, we must value the incremental costs and benefits associated with that de- cision. A good decision is one for which the value of the benefits exceeds the value of the costs. To compare costs and benefits that occur at different points in time, in different currencies, or with different risks, we must put all costs and benefits in common terms. Typically, we convert costs and benefits into cash today. A competitive market is one in which a good can be bought and sold at the same price. We use prices from competitive markets to determine the cash value of a good. 3.2 Interest Rates and the Time Value of Money The time value of money is the difference in value between money today and money in the future. The rate at which we can exchange money today for money in the future by borrowing or investing is the current market interest rate. The risk-free interest rate, r f , is the rate at which money can be borrowed or lent without risk. 3.3 Present Value and the NPV Decision Rule The present value (PV) of a cash flow is its value in terms of cash today. The net present value (NPV) of a project is PV (Benefits) − PV (Costs)  (3.1) A good project is one with a positive net present value. The NPV Decision Rule states that when choosing from among a set of alternatives, choose the one with the highest NPV. The NPV of a project is equivalent to the cash value today of the project. Regardless of our preferences for cash today versus cash in the future, we should always first maximize NPV. We can then borrow or lend to shift cash flows through time and to find our most preferred pattern of cash flows. 3.4 Arbitrage and the Law of One Price Arbitrage is the process of trading to take advantage of equivalent goods that have different prices in different competitive markets. A normal market is a competitive market with no arbitrage opportunities. The Law of One Price states that if equivalent goods or securities trade simultaneously in dif- ferent competitive markets, they will trade for the same price in each market. This law is equiva- lent to saying that no arbitrage opportunities should exist. 3.5 No-Arbitrage and Security Prices The No-Arbitrage Price of a Security is PV ( All cash flows paid by the security )  (3.3) No-arbitrage implies that all risk-free investments should offer the same return. The Separation Principle states that security transactions in a normal market neither create nor destroy value on their own. As a consequence, we can evaluate the NPV of an investment deci- sion separately from the security transactions the firm is considering. To maximize the value of the entire firm, managers should make decisions that maximize the NPV. The NPV of the decision represents its contribution to the overall value of the firm. Value additivity implies that the value of a portfolio is equal to the sum of the values of its parts. M03_BERK6318_06_GE_C03.indd 120 27/04/23 10:51 AM  Problems 121 Key Terms arbitrage p. 110 normal market p. 111 arbitrage opportunity p. 110 NPV Decision Rule p. 107 bond p. 112 portfolio p. 116 competitive market p. 101 present value (PV) p. 105 discount factor p. 105 return p. 114 discount rate p. 105 risk-free interest rate p. 104 financial security p. 111 security p. 111 future value (FV) p. 105 Separation Principle p. 116 interest rate factor p. 104 short sale p. 113 Law of One Price p. 111 time value of money p. 103 net present value (NPV) p. 106 Valuation Principle p. 102 no-arbitrage price p. 113 value additivity p. 117 Further Many of the fundamental principles of this chapter were developed in the classic text by I. Fisher, Reading The Theory of Interest: As Determined by Impatience to Spend Income and Opportunity to Invest It (Macmillan, 1930); reprinted (Augustus M. Kelley, 1955). To learn more about the principle of no arbitrage and its importance as the foundation for modern finance theory, see S. Ross, Neoclassical Finance (Princeton University Press, 2004). For a discussion of arbitrage and rational trading and their role in determining market prices, see M. Rubinstein, “Rational Markets: Yes or No? The Affirmative Case,” Financial Analysts Journal 57 (2001): 15–29. For a discussion of some of the limitations to arbitrage that may arise in practice, see A. Shleifer and R. Vishny, “Limits of Arbitrage,” Journal of Finance 52 (1997): 35–55 and A. Shleifer, Inefficient markets: An introduction to behavioural finance. OUP Oxford, 2000. Problems Valuing Decisions 1. Motor Company is considering offering a $1,600 rebate on its minivan, lowering the vehicle’s price from $29,000 to $27,400. The marketing group estimates that this rebate will increase sales over the next year from 42,000 to 60,000 vehicles. Suppose Honda’s profit margin with the rebate is $5400 per vehicle. If the change in sales is the only consequence of this decision, what are its costs and benefits? Is it a good idea? 2. You are an international shrimp trader. A food producer in the Czech Republic offers to pay you 2.9 million Czech koruna today in exchange for a year’s supply of frozen shrimp. Your Thai supplier will provide you with the same supply for 2.2 million Thai baht today. If the current competitive market exchange rates are 24.24 koruna per dollar and 37.74 baht per dollar, what is the value of this deal? 3. Suppose the current market price of corn is $3.75 per bushel. Your firm has a technology that can convert 1 bushel of corn to 3 gallons of ethanol. If the cost of conversion is $1.84 per bushel, at what market price of ethanol does conversion become attractive? The minimum price in which ethanol becomes attractive is ($3.75 + $1.84 / bushel of corn) / (3 gallons of ethanol / bushel of corn) = $1.86 per gallon of ethanol. 4. Suppose your employer offers you a choice between a $6900 bonus and 400 shares of the com- pany stock. Whichever one you choose will be awarded today. The stock is currently trading for $52 per share. a. Suppose that if you receive the stock bonus, you are free to trade it. Which form of the bonus should you choose? What is its value? M03_BERK6318_06_GE_C03.indd 121 27/04/23 10:51 AM  122 Chapter 3 Financial Decision Making and the Law of One Price b. Suppose that if you receive the stock bonus, you are required to hold it for at least one year. What can you say about the value of the stock bonus now? What will your decision depend on? 5. You have decided to take your daughter skiing in Nagano, Japan. The best price you have been able to find for a roundtrip air ticket is $555. You notice that you have 20,000 frequent flier miles that are about to expire, but you need 25,000 miles to get her a free ticket. The airline offers to sell you 5000 additional miles for $0.05 per mile. a. Suppose that if you don’t use the miles for your daughter’s ticket they will become worthless. What should you do? b. What additional information would your decision depend on if the miles were not expiring? Why? Interest Rates and the Time Value of Money 6. Suppose the risk-free interest rate is 3.2%. a. Having $500 today is equivalent to having what amount in one year? b. Having $500 in one year is equivalent to having what amount today? c. Which would you prefer, $500 today or $500 in one year? Does your answer depend on when you need the money? Why or why not? 7. You have an investment opportunity in Japan. It requires an investment of $0.98 million today and will produce a cash flow of ¥107 million in one year with no risk. Suppose the risk-free interest rate in the United States is 3.9%, the risk-free interest rate in Japan is 2.3%, and the current competitive exchange rate is ¥110 per $1. What is the NPV of this investment? Is it a good opportunity? 8. Your firm has a risk-free investment opportunity where it can invest $163,000 today and receive $179,000 in one year. For what level of interest rates is this project attractive? (Round the inter- est rate to three digits after decimal.) Present Value and the NPV Decision Rule 9. You run a construction firm. You have just won a contract to construct a government office building. It will take one year t