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a.Which investment is most valuable?

b.Suppose each investment would require use of the same parcel of land. Therefore you can take only one. Which one? Hint: What is the firm’s objective: to earn a high rate of return or to increase firm value?

5.In Section 2.1, we analyzed the possible construction of an office building on a plot of land appraised at $50,000. We concluded that this investment had a positive NPV of $7,143 at a discount rate of 12 percent.

Suppose E. Coli Associates, a firm of genetic engineers, offers to purchase the land for $60,000, $30,000 paid immediately and $30,000 after one year. United States government securities maturing in one year yield 7 percent.

a.Assume E. Coli is sure to pay the second $30,000 installment. Should you take its offer or start on the office building? Explain.

b.Suppose you are not sure E. Coli will pay. You observe that other investors demand a 10 percent return on their loans to E. Coli. Assume that the other investors have correctly assessed the risks that E. Coli will not be able to pay. Should you accept E. Coli’s offer?

6.Explain why the discount rate equals the opportunity cost of capital.

7.Norman Gerrymander has just received a $2 million bequest. How should he invest it?

8.Show that your answers to Practice Question 7 are consistent with the rate of return rule for investment decisions.

9.Take another look at investment opportunity (d) in Practice Question 7. Suppose a bank offers Norman a $600,000 personal loan at 8 percent. (Norman is a long-time customer of the bank and has an excellent credit history.) Suppose Norman borrows the money, invests $1 million in real estate opportunity (d) and puts the rest of his money in opportunity (c), the stock market. Is this a smart move? Explain.

10.Respond to the following comments.

a.“My company’s cost of capital is the rate we pay to the bank when we borrow money.”

b.“Net present value is just theory. It has no practical relevance. We maximize profits. That’s what shareholders really want.”

c.“It’s no good just telling me to maximize my stock price. I can easily take a short view and maximize today’s price. What I would prefer is to keep it on a gently rising trend.”

11.Ms. Smith is retired and depends on her investments for retirement income. Mr. Jones is a young executive who wants to save for the future. They are both stockholders in Airbus, which is investing over $12 billion to develop the A380, a new super-jumbo airliner. This investment’s payoff is many years in the future. Assume the investment is positive-NPV for Mr. Jones. Explain why it should also be positive-NPV for Ms. Smith.

12.Answer this question by drawing graphs like Figure 2.1. Casper Milktoast has $200,000 available to support consumption in periods 0 (now) and 1 (next year). He

3.Draw a figure like Figure 2.1 to represent the following situation.

a.A firm starts out with $10 million in cash.

b.The rate of interest r is 10 percent.

c.To maximize NPV the firm invests today $6 million in real assets. This leaves $4 million which can be paid out to the shareholders.

d.The NPV of the investment is $2 million.

When you have finished, answer the following questions:

e.How much cash is the firm going to receive in year 1 from its investment?

f.What is the marginal return from the firm’s investment?

g.What is the PV of the shareholders’ investment after the firm has announced its investment plan?

h.Suppose shareholders want to spend $6 million today. How can they do this?

i.How much will they then have to spend next year? Show this on your drawing.

4.For an outlay of $8 million you can purchase a tanker load of bucolic acid delivered in Rotterdam one year hence. Unfortunately the net cash flow from selling the tanker load will be very sensitive to the growth rate of the world economy:

a.What is the expected cash flow? Assume the three outcomes for the economy are equally likely.

b.What is the expected rate of return on the investment in the project?

c.One share of stock Z is selling for $10. The stock has the following payoffs after one year:

Calculate the expected rate of return offered by stock Z. Explain why this is the opportunity cost of capital for your bucolic acid project.

d. Calculate the project’s NPV. Is the project a good investment? Explain why.

5.In real life the future health of the economy cannot be reduced to three equally probable states like slump, normal, and boom. But we’ll keep that simplification for one more

You have identified the possible payoffs to investors in three stocks, X, Y, and Z:

a.What are the expected cash inflows of projects B and C?

b.What are the expected rates of return offered by stocks X, Y, and Z?

c.What are the opportunity costs of capital for projects B and C? Hint: Calculate the percentage differences, slump versus normal and boom versus normal, for stocks X, Y, and Z. Match up to the percentage differences in B’s and C’s payoffs.

d.What are the NPVs of projects B and C?

e.Suppose B and C are launched and $5 million is invested in each. How much will they add to the total market value of your company’s shares?

Valuing Cash Flows in Several Periods

One of the nice things about present values is that they are all expressed in current dollars—so that you can add them up. In other words, the present value of cash flow A B is equal to the present value of cash flow A plus the present value of cash flow B. This happy result has important implications for investments that produce cash flows in several periods.

We calculated above the value of an asset that produces a cash flow of C1 in year 1, and we calculated the value of another asset that produces a cash flow of C2 in year 2. Following our additivity rule, we can write down the value of an asset that produces cash flows in each year. It is simply

We can obviously continue in this way to find the present value of an extended stream of cash flows:

This is called the discounted cash flow (or DCF) formula. A shorthand way to write it is

Ct

PV a 11 rt 2t

where refers to the sum of the series. To find the net present value (NPV) we add the (usually negative) initial cash flow, just as in Chapter 2:

Ct

NPV C0 PV C0 a 11 rt 2t

Why the Discount Factor Declines as Futurity Increases—

And a Digression on Money Machines

If a dollar tomorrow is worth less than a dollar today, one might suspect that a dollar the day after tomorrow should be worth even less. In other words, the discount factor DF2 should be less than the discount factor DF1. But is this necessarily so, when there is a different interest rate rt for each period?

Suppose r1 is 20 percent and r2 is 7 percent. Then

DF1 1.201 .83

1

DF2 11.07 22 .87

Apparently the dollar received the day after tomorrow is not necessarily worth less than the dollar received tomorrow.

But there is something wrong with this example. Anyone who could borrow and lend at these interest rates could become a millionaire overnight. Let us see how such a “money machine” would work. Suppose the first person to spot the opportunity is Hermione Kraft. Ms. Kraft first lends $1,000 for one year at 20 percent. That is an attractive enough return, but she notices that there is a way to earn

an immediate profit on her investment and be ready to play the game again. She reasons as follows. Next year she will have $1,200 which can be reinvested for a further year. Although she does not know what interest rates will be at that time, she does know that she can always put the money in a checking account and be sure of having $1,200 at the end of year 2. Her next step, therefore, is to go to her bank and borrow the present value of this $1,200. At 7 percent interest this present value is

1200

PV 11.07 22 $1,048

Thus Ms. Kraft invests $1,000, borrows back $1,048, and walks away with a profit of $48. If that does not sound like very much, remember that the game can be played again immediately, this time with $1,048. In fact it would take Ms. Kraft only 147 plays to become a millionaire (before taxes).1

Of course this story is completely fanciful. Such an opportunity would not last long in capital markets like ours. Any bank that would allow you to lend for one year at 20 percent and borrow for two years at 7 percent would soon be wiped out by a rush of small investors hoping to become millionaires and a rush of millionaires hoping to become billionaires. There are, however, two lessons to our story. The first is that a dollar tomorrow cannot be worth less than a dollar the day after tomorrow. In other words, the value of a dollar received at the end of one year (DF1) must be greater than the value of a dollar received at the end of two years (DF2). There must be some extra gain2 from lending for two periods rather than one: (1 r2)2 must be greater than 1 r1.

Our second lesson is a more general one and can be summed up by the precept “There is no such thing as a money machine.”3 In well-functioning capital markets, any potential money machine will be eliminated almost instantaneously by investors who try to take advantage of it. Therefore, beware of self-styled experts who offer you a chance to participate in a sure thing.

Later in the book we will invoke the absence of money machines to prove several useful properties about security prices. That is, we will make statements like “The prices of securities X and Y must be in the following relationship—otherwise there would be a money machine and capital markets would not be in equilibrium.”

Ruling out money machines does not require that interest rates be the same for each future period. This relationship between the interest rate and the maturity of the cash flow is called the term structure of interest rates. We are going to look at term structure in Chapter 24, but for now we will finesse the issue by assuming that the term structure is “flat”—in other words, the interest rate is the same regardless of the date of the cash flow. This means that we can replace the series of interest rates r1, r2, . . . , rt, etc., with a single rate r and that we can write the present value formula as

1That is, 1,000 (1.04813)147 $1,002,000.

2The extra return for lending two years rather than one is often referred to as a forward rate of return. Our rule says that the forward rate cannot be negative.

3The technical term for money machine is arbitrage. There are no opportunities for arbitrage in wellfunctioning capital markets.

Calculating PVs and NPVs

You have some bad news about your office building venture (the one described at the start of Chapter 2). The contractor says that construction will take two years instead of one and requests payment on the following schedule:

1.A $100,000 down payment now. (Note that the land, worth $50,000, must also be committed now.)

2.A $100,000 progress payment after one year.

3.A final payment of $100,000 when the building is ready for occupancy at the end of the second year.

Your real estate adviser maintains that despite the delay the building will be worth $400,000 when completed.

All this yields a new set of cash-flow forecasts:

150,000 100,000 300,000

1.0711.07 22

Table 3.1 calculates NPV step by step. The calculations require just a few keystrokes on an electronic calculator. Real problems can be much more complicated, however, so financial managers usually turn to calculators especially programmed for present value calculations or to spreadsheet programs on personal computers. In some cases it can be convenient to look up discount factors in present value tables like Appendix Table 1 at the end of this book.

Fortunately the news about your office venture is not all bad. The contractor is willing to accept a delayed payment; this means that the present value of the contractor’s fee is less than before. This partly offsets the delay in the payoff. As Table 3.1 shows,