Finance 100: Corporate Finance Professor Michael R. Roberts Quiz 3 November 16, 2005 Name: Section: Question Maximum Student Score 1 40 2 35 3 25 Total 100 Instructions: Please read each question carefully Formula sheets are attached to the back of the quiz You must show all work to receive credit & all numbers must be precise to the 100 th place (i.e., 0.01). Good luck 1
1. (40 Points) You are given the following information about possible investments: Expected Standard Correlation with Asset Return (E(R)) Deviation (σ) the market (ρ) Market Portfolio 12% 20%?? Small Stocks?? 40% 1 Large Stocks?? 30% 0.5 Silver?? 20% -0.5 T-Bills 6% 0%?? (a) (12 Points) What are the CAPM betas of small stocks, large stocks, silver and T-bills? Solution: Small Stock = 1 0.40 0.20 = 2 Large Stock = 0.5 0.30 0.20 = 0.75 Silver = 0.5 0.20 0.20 = 0.5 T-Bills = 0 0.00 0.20 = 0 2
(b) (12 Points) Assume that all assets are priced correctly according to the CAPM. What are the expected returns to small stocks, large stocks, silver, and T-bills? Solution: textsmallstock = 0.06 + 2(0.06) = 0.18 textlargestock = 0.06 + 0.75(0.06) = 0.105 textsilver = 0.06 0.5(0.06) = 0.03 textt Bills = 0.06 + 0(0.06) = 0.06 (c) (6 Points) Explain VERY BRIEFLY the relation between the expected return to silver (a risky asset) and T-Bills (a riskless assets) that you computed above in part (b). Solution: Silver has a lower expected return that T-bills because it acts as a hedge against market risk and therefore is relatively expensive. (Idiosyncratic risk does not matter in a CAPM world. ) 3
(d) (5 Points) In addition to the assets described above you are told that the expected return on NewFirm Inc. shares is 25% and that its beta is 3.0. Has the market correctly priced this firm s stock, according to the CAPM? If not, explain whether it is over or under valued. Solution: The CAPM predicts NewFirm s expected return should be: textnewf irm = 0.06 + 3(0.06) = 0.24 Therefore, the stock is undervalued today (i.e., its expected return is to high). (e) (5 Points) If you found that NewFirm is mispriced, how would you take advantage of this mispricing using portfolios on the Capital Market Line? Is this an arbitrage opportunity? Solution: Start by buying NewFirm, which is undervalued in the market. Then use the CML to find a portfolio consisting of the market and risk-free such beta is the same as NewFirm s. Sell this portfolio. Matching the beta of the portfolio implies the weight on the market is 3 and the weight on the risk-free is -2. The gain in expected return is 1%. This is NOT an arbitrage opportunity but rather a strategy that will work on average, assuming that the CAPM is correct. 4
2. (35 Points) On May 20 you observe that the current S&P 500 index is 1034.21 while the June index futures price on the index is 1037. Assume that the riskless rate of interest is 5% per annum (continuously compounded) and that the June futures contract will expire in exactly one month. (a) (8 Points) Suppose you buy two stock index futures. What is the payoff to your position if the S&P 500 index closes at 1,050? What is the payoff if the index closes at 1,000? (you may assume that one index point is worth $250.). Solution: Your payoffs are: Close = 1, 050 = P ayoff = 2 $250 (1, 050 1, 037) = $6, 500 Close = 1, 000 = P ayoff = 2 $250 (1, 000 1, 037) = $18, 500 5
(b) (16 Points) Assume that the dividend yield is 2.92% (continuously compounded). Is the June futures contract correctly priced? If not, explain in detail what you do in order to profit from this mispricing. Be sure to clearly state the transactions and all corresponding cash flows. Solution: The theoretical futures price is 1034.21exp{(0.05 0.02921)/12} = 1, 036 which is $1 less than the market price. The arbitrage table is: Time Period (Months) Position 0 1 Short Future 0 1,037 - S Buy Index -1,034.21 exp{-0.02921/12} = -1,031.70 S Short Bond (Borrow) 1,037 exp{-0.05/12} = 1,032.69-1,037 Net Cash Flows 0.99 0 6
(c) (6 Points) What is the dividend yield implied by the futures price in the market? Solution: No arbitrage implies: so the implied dividend yield, d, is: F = S 0 exp{r d)t } d = 0.05 12 ln(1, 037/1, 034.21) = 0.0177 (d) (5 Points) Explain what would happen to the June index futures if the market suddenly expects the dividend yield on the index to decrease starting July 1 of that same year. Solution: Lower dividends on the index would send the price of the index (S 0 ) down. This would reduce the June futures price. This change, however, is irrelevant for d in the contract since it occurs after the contract expiration. 7
3. (25 Points) The president of your company has to make a choice between two investments: Internal Rate Project CF 0 CF 1 CF 2 of Return A -400 +241 +293 21% B -200 +131 +172 31% The opportunity cost of capital is 9% and the president wants to invest in project B because it offers a higher IRR. (You can assume annual compounding.) (a) (12 Points) How would you explain to the president that he is making an incorrect decision? Solution: He should look at the NPV for each project and pick the larger of the two. The NPVs for the two projects are: NP V (A) = 400 + 241(0.9174) + 293(0.8417) = 67.71 NP V (B) = 200 + 131(0.9174) + 172(0.8417) = 64.95 Thus, he should choose project A, not B. 8
(b) (6 Points) Why does the president s criterion lead him to the wrong conclusion? In other words, explain briefly the intuition behind why the IRR criterion, as used by the president, leads to an incorrect choice. Solution: The IRR does not take into account the relative scale of mutually exclusive projects. Therefore, a large scale project that may contribute more to firm value will be rejected because of its high initial investment. (c) (7 Points) Can you show the president how to adapt the IRR rule to choose the best project? (While you don t need to compute the IRR, set up the problem and explain what you would do.) Solution: You would have to find the IRR of the increment. For example, consider the cash flows to project A minus project B: Solve for the IRR of Internal Rate Project CF 0 CF 1 CF 2 of Return A - B -200 110 121 these incremental cash flows: 200 + 110 1 + IRR + 121 (1 + IRR) 2 = 0 We get IRR = 10% > 9% so we would accept this project. 9