# Problem Set 4: Capital Budgeting and Other Topics: Problem Set

### (Due before section, Monday, October 28)

B.A. 130

1. "If the efficient market hypothesis is true, then a portfolio manager might just as well select his or her portfolio by throwing darts at an open Wall Street Journal." Is this statement correct? Why or why not?

• Two reasons why it isn't completely true. First, throwing darts isn't guaranteed to minimize idiosyncratic risk--and a well-chosen, well-diversified portfolio will minimize idiosyncratic risk. Since you get no extra expected return from bearing idiosyncratic risk, it is not worth having any of it in your portfolio. Second, throwing darts won't produce a portfolio with the particular beta that you wish. It is important to pick a portfolio of the proper beta--if not, you are either bearing too much risk (and the extra returns that you expect are not enough to compensate), or you are bearing too little risk.

2. Suppose a company announces that it is going to split its stock two for one. Would you expect its stock to rise? Why or why not?

• Probably it would rise. Investors are likely to take the split announcement as a sign that good earnings and dividend news is on the way. If earnings and dividends do not rise subsequently, expect the stock to give back its gains.

3. Consider an asset with the following cash flows:

 Year Zero Year One Year Two Year Three Cash Flows: -\$60 +\$26 +\$24 +\$22

The discount rate is 10% per year. Suppose that the firm uses straight-line book-value depreciation. Calculate the book rate of return in each year. Calculate economic depreciation. Calculate the true profitability of the asset. Account for any difference between book returns and economic profitability.

• This is a zero NPV investment at 10% per year. To see this, start from year three and work backward. The year-three cash flow of \$22 has a value of \$40 in year two--so as of year two the asset is worth \$44. The \$44 value as of year two is worth \$40 in year one--so as of year one the asset is worth \$66. And \$66 in year one is worth +\$60 in year zero, exactly balancing out the purchase price of the asset. Look at how the value of the asset changes each year, and see that its value shrinks by \$20 a year. The amount by which its value shrinks is "economic" depreciation. Thus here "economic" depreciation is identical to straight-line "book" depreciation. Thus (as Brealey and Myers demonstrated) there is no difference between "book" returns and "economic" profitability.

4. What is "economic depreciation"? Why would accountants do a better job if they used depreciation schedules that matched expected economic depreciation?

• Economic depreciation is the change (usually the decline) in the present value of an asset. We finance economists think that accountants would do a better job if they adjusted their depreciation schedules to reflect "economic" rather than "accounting" depreciation because economic depreciation better captures the opportunity cost of employing the asset.

5. For what kinds of capital investment projects are Monte Carlo analyses most useful?

• Those in which there is substantial dependence between the different factors--so that sales are low when costs are high, and so forth. When the things that could affect the value all vary together, looking at the sensitivity of present-value calculations by varying factors one at a time will understate uncertainty.

6. For what kinds of capital investment projects are decision tree analyses most useful?

• Those in which the company has further decisions to make, after the initial investment is undertaken, that can materially affect the investment's profitability.

7. Suppose that the Amalgamated Chemical Company is engaged in a highly competitive segment of the world chemical industry--a segment so risky that the required rate of return is 20% per year. Suppose that the accounting information for a standard plant with a three-year life built by Amalgamated Chemical (or one of its competitors) is as below (with all numbers in millions):

 Year Zero One Two Three Initial Investment -\$100 Revenues \$100 \$100 \$100 Operating Costs -\$50 -\$50 -\$50 Salvage Value +\$25 Depreciation -\$33.33 -\$33.33 -\$33.33 Pretax Income \$16.67 \$16.67 \$41.67 Tax at 40% \$6.67 \$6.67 \$16.67 Net After-Tax Income \$10 \$10 \$25 Cash Flow -\$100 +\$43.33 +\$43.33 +\$58.33

a. Calculate the net present value of investing in a plant.

• The net present value of investing in an old plant with the old technology is zero (at 20% discount rate): this makes sense, because it is a competitive industry. If it were a competitive industry and someone claimed that investing in a new plant was a positive-NPV investment, you should be highly suspicious.

b. Calculate the value of a one year old plant and of a two year old plant.

• two-year old plant: = cash flow received in year three discounted back to year two = \$58.33/1.2 = \$48.6
• one-year old plant: = cash flows received in years two and three discounted back to year one = \$76.6

c. Suppose that technological breakthroughs create a new kind of plant--this new improved technology requires only half as large an initial investment (and half as large a salvage value) and produces the same output with only half the operating cost. This new technology is freely available to every firm producing in this competitive industry. What do you expect the annual revenues from operating a plant--either a new technology plant, or an old technology plant--to be after this technological breakthrough?

• Expect revenues to fall in half--to \$50 a year. It is still a competitive industry; people will build new plants until revenues fall low enough to make it no longer worth their while. Because all costs have fallen in half as a result of the new technology, you need only half the revenue--\$50 a year--to make the plant a zero-NPV investment. And people will build plants until the revenue from each plant is driven down to \$50 a year, and building the next new plant does become a zero-NPV investment.

d. What is the net present value of investing in a new technology plant? In an old technology plant after the technological breakthrough?

• The NPV of a new technology plant will be zero (after all, you have no competitive advantage to create economic rents; how then in a competitive industry can you expect to make positive NPV returns from an investment tht everyone else can duplicate?). The NPV of an old technology plant would be (if operated for its entire lifetime) -\$91.3

e. Calculate the post-breakthrough value of a one year old old-technology plant and of a two year old old-technology plant.

• \$8.68 for a one-year plant; \$10.42 for a two-year plant (if both are operated until scrapping time)

f. Suppose that you could scrap a two year old plant and receive \$25 million in salvage value (as opposed to waiting a year, producing in year three, and then scrapping the plant for \$12.5 million in salvage value). Which is the best alternative?

• Scrap now. The plant is barely covering its operating costs. Waiting not only loses money through discounting, but as the salvage value falls as well.
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