## Problem Set # 2

1. Consider two stocks, of equal systematic riskiness:
• Stock A is expected to pay a dividend of \$10 a share forever.
• Stock B is expected to pay a dividend of \$5 a year next year, and thereafter each year's dividend is expected to be 5% above the previous year's dividend.
At what required rate of return r would the market price of these two stocks be equal? What would the value of these two stocks be at that required rate of return?

2. Suppose that we have two stocks, each of which will pay out \$5 per share next year in dividends, and report \$10 per share in earnings. Suppose that the required rate of return is 10%, that one of the stocks--stock C--has a price/earnings ratio of 10, and that the other stock--stock D--has a price/earnings ratio of 20.
• Explain how two stocks with identical earnings and dividends this year can nevertheless sell for different prices.
• What is the expected long-run dividend growth rate g for stock C? For stock D?
• What is the PVGO--the present value of growth opportunities--for stock C? For stock D?
3. Suppose that you are playing a card game (using a normal deck of 52 cards, with ace high; and with spades higher than hearts higher than diamonds higher than clubs) with one other person: each is dealt a card which he or she does not examine, but instead shows to his or her opponent. After a round of negotiation and discussion, the cards are turned over, and the low card holder pays the high card holder \$1. You look at your opponent's card: it is the eight of diamonds.
• What--assuming that you dealt, and did not cheat--is your expected gain (or loss) from this hand?
• What is the variance of your return from this hand? What is the standard deviation?
• Your opponent offers you \$0.25 if you will cancel this hand and deal another one. Should you accept or reject his or her offer? Why or why not?
4. Suppose that you are playing a dice game (using two normal, fair, six sided dice). If you roll a seven using the two dice, your opponent pays you \$1; if you roll an 11, you pay your opponent \$1; if you roll anything else you roll again until you do roll a 7 or an 11.
• What is your expected gain (or loss) from this game?
• What is the variance of your return from this game? What is the standard deviation?
• Your opponent offers you \$0.80 if you will cancel this game. Should you accept or reject his or her offer? Why or why not?
5. Mr. Cyrus Clops has to make a choice between two possible investments:

A
 Project C(0) C(1) C(2) Project -500 480 144 Project B -100 65 84.5

• What are the internal rates of return (IRR) associated with the two projects?
• If you can undertake only one of them, and if the cost of capital is 10%, which should you undertake?
• Explain why IRR analysis gives the same (or different) answer as net present value analysis.
• Suppose the cost of capital is 20%; which should you undertake?
6. Suppose that you have to choose between two machines which do the same job but have different lives. You must buy one of the two machines. The two machines have the following costs, measured in real, inflation-adjusted dollars:

A
 Year Machine Machine B 0 \$20,000 \$25,000 1 \$5,000 \$4,000 2 \$5,000 + replace \$4,000 3 \$4,000+replace
Suppose your cost of capital is 10% per year.

Which machine should you buy? Why? What assumptions are you making about what happens in year 2 (or 3) when you have to replace the machine bought in year 0?

7. What was the average gap between the annual returns on a diversified portfolio of small stocks and the returns on a diversified portfolio of U.S. Treasury bonds between 1926 and 1994? Why does anyone invest in Treasury bonds at all--what advantage do investments in Treasury bonds have over investments in the stocks of small firms?

8. You believe that there is a 50% chance that stock E will rise by 20% and a 50% chance that stock E will fall by 10%. You also believe that there is a 1/3 chance that stock F will rise by 30% and a 2/3 chance that stock F will remain constant. The correlation coefficient between the two stocks is .25
• Calculate the expected return, the variance, and the standard deviation of each stock.
• Calculate the expected return, the variance, and the standard deviation of a portfolio made up of equal investments in each stock.
9. Suppose that you are trying to calculate the variance of a portfolio made up of all the stocks in the S&P 500 index--500 stocks--from knowledge of the variances and covariances of the stocks alone.
• How many variance terms are there in your calculation?
• How many covariance terms?
• Suppose that each of the stocks has a standard deviation of annual returns of 25% and a correlation with each other stock of .5. What is the standard deviation of a portfolio that places 1/500 of your wealth in each of the 500 stocks?
• What is the standard deviation of a portfolio that places all of your wealth in one of the stocks?
10. Brealey and Myers say that diversification is of the utmost importance for investors making portfolio decisions. Is diversification important for businesses making investment decisions? Why or why not?