JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Homework #5 Suggested Solutions Problem 1. (9.4) Define the following terms: (a) Cost of debt (b) Cost of equity (c) After tax WACC (d) Equity beta (e) Asset beta (f) Pure play comparable (g) Certainty equivalent (a) The expected return on debt. If the debt has very low default risk, this is close to its yield to maturity. (b) The expected return on equity. (c) A weighted average of the cost of equity and the after tax cost of debt, where the weights are the relative market values of the firm s debt and equity. (d) The change in the return of the stock for each additional 1% change in the market return. (e) The change in the return on a portfolio of all the firm s securities (debt and equity) for each additional 1% change in the market return. (f) A company specializing in one activity that is similar to that of a division of a more diversified company. (g) A certain cash flow occurring at time t with the same present value as an uncertain cash flow at time t. Problem 2. (9.10) A project has a forecasted cash flow of $110 in year 1 and $121 in year 2. The interest rate is 5%, the estimated risk premium on the market is 10%, and the project has a beta of 0.5. If you use a constant risk adjusted discount rate, what is (a) The PV of the project? (b) The certainty equivalent cash flow in year 1 and year 2? (c) The ratio of the certainty equivalent cash flows to the expected cash flows in years 1 and 2? (a) P V = 110 1 + r f + β(r m r f ) + 121 (1 + r f + β(r m r f )) 2 = 110 1.1 + 121 1.1 2 = $200. 1
(b) CEQ 1 1.05 = 110 1.1 CEQ 1 = $105; CEQ 2 = 121 1.05 2 1.1 CEQ 2 2 = $110.25. (c) Ratio 1 = 105 110 = 0.95; Ratio 2 = 110.25 = 0.91. 121 Problem 3. (9.11) The total market value of the common stock of the Okefenokee Real Estate Company is $6 million, and the total value of its debt is $4 million. The treasurer estimates that the beta of the stock is currently 1.5 and that the expected risk premium on the market is 6%. The Treasury bill rate is 4%. Assume for simplicity that Okefenokee debt is risk free and the company does not pay tax. (a) What is the required return on Okefenokee stock? (b) Estimate the company cost of capital. (c) What is the discount rate for an expansion of the company s present business? (d) Suppose the company wants to diversify into the manufacture of rose colored spectacles. The beta of unleveraged optical manufacturers is 1.2. Estimate the required return on Okefenokee s new venture. (a) r equity = r f + β(r m r f ) = 0.04 + 1.5 0.06 = 0.13 or 13%. (b) r assets = D V r debt + E V r $4 million $6 million equity = 0.04 + 0.13 = 0.094 or 9.4%. $10 million $10 million (c) The cost of capital depends on the risk of the project being evaluated. If the risk of the project is similar to the risk of the other assets of the company, then the appropriate rate of return is the company cost of capital. Here, the appropriate discount rate is 9.4%. (d) r equity = r f + β(r m r f ) = 0.04 + 1.2 0.06 = 0.112 or 11.2%. r assets = D V r debt+ E V r $4 million $6 million equity = 0.04+ 0.112 = 0.0832 or 8.32%. $10 million $10 million Problem 4. (9.12) Nero Violins has the following capital structure: (a) What is the firm s asset beta? (Hint: What is the beta of a portfolio of all the firm s securities?) (b) Assume that the CAPM is correct. What discount rate should Nero set for investments that expand the scale of its operations without changing its asset beta? Assume a risk free interest rate of 5% and a market risk premium of 6%. D (a) β assets = β debt V + β P preferred V + β C common V = 2
$100 million = 0 $439 million + 0.2 $40 million $439 million + 1.2 $299 million $439 million = 0.836. (b) r = r f + β(r m r f ) = 0.05 + 0.836 0.06 = 0.10016 or 10.016%. Problem 5. (20.1) Complete the following passage: A option gives its owner the opportunity to buy a stock at a specified price that is generally called the price. A option gives its owner the opportunity to sell stock at a specified price. Options that can be exercised only at maturity are called options. Call; exercise; put; European. Problem 6. (20.3) Suppose that you hold a share of stock and a put option on that share. What is the payoff when the option expires if (a) the stock price is below the exercise price? (b) the stock price is above the exercise price? (a) The exercise price of the put option (i.e., you d sell stock for the exercise price). (b) The value of the stock (i.e., you would throw away the put and keep the stock). Problem 7. (20.9) What is a call option worth if (a) the stock price is zero? (b) the stock price is extremely high relative to the exercise price? (a) Zero. (b) Stock price less the present value of the exercise price. Problem 8. (20.10) How does the price of a call option respond to the following changes, other things equal? Does the call price go up or down? (a) Stock price increases. (b) Exercise price is increased. (c) Risk free rate increases. (d) Expiration date of the option is extended. (e) Volatility of the stock price falls. (f) Time passes, so the option s expiration date comes closer. The call price (a) increases; (b) decreases; (c) increases; (d) increases; 3
(e) decreases; (f) decreases. Problem 9. (20.15) It is possible to buy three month call options and three month puts on stock Q. Both options have an exercise price of $60 and both are worth $10. If the interest rate is 5% a year, what is the stock price? (Hint: Use put call parity.) Let P 3 = the value of the three month put, C 3 = the value of the three month call, S = the market value of a share of stock, and EX = the exercise price of the options. Then, from put call parity: V alue of call + P resent value of exercise price = V alue of put + Share price, C 3 + EX (1 + r) 0.25 = P 3 + S. Since both options have an exercise price of $60 and both are worth $10, then: S = EX $60 = = $59.27. (1 + r) 0.25 1.050.25 Problem 10. (20.22) The common stock of Triangular File Company is selling at $90. A 26 week call option written on Triangular File s stock is selling for $8. The call s exercise price is $100. The risk free interest rate is 10% per year. (a) Suppose that puts on Triangular stock are not traded, but you want to buy one. How would you do it? (b) Suppose that puts are traded. What should a 26 week put with an exercise price of $100 sell for? (a) Use the put call parity relationship for European options: V alue of call + P resent value of exercise price = V alue of put + Share price. Solve for the value of the put: V alue of put = V alue of call + P V (EX) Share price. Thus, to replicate the payoffs for the put, you would buy a 26 week call with an exercise price of $100, invest the present value of the exercise price in a 26 week risk free security, and sell the stock short. (b) Using the put call parity relationship, the European put will sell for: Problem 11. (21.2) $8 + $100 1.05 $90 = $13.24. (a) Can the delta of a call option be greater than 1? Explain. 4
(b) Can it be less than zero? (c) How does the delta of a call change if the stock price rises? (d) How does it change if the risk of the stock increases? (a) No. The maximum delta is 1 when the ratio of stock price to exercise price is very high. (b) No. (c) Delta increases. (d) Delta increases. Problem 12. (21.5) Over the coming year Ragwort s stock price will halve to $50 from its current level of $100 or it will rise to $200. The one year interest rate is 10%. (a) In a risk neutral world what is the probability that Ragwort stock will rise in price? (b) Use the risk neutral method to value a one year call option on Ragwort stock with an exercise price of $100. (c) What is the option delta? (d) Use the replicating portfolio method to value this call. (e) If someone told you that in reality there is a 60% chance that Ragwort s stock price will rise to $200, would you change your view about the value of the option? Explain. (a) p 100 + (1 p) ( 50) = 10 p = 0.4. 0.4 100 + 0.6 0 (b) V alue of call = = $36.36. 1.1 spread of option prices (c) Delta = spread of stock prices = 100 0 200 50 = 0.667. (d) The replicating portfolio is the following: Current Possible Future Cash Flow Cash Flows Buy call $36.36 0 $100 equals Buy 0.667 shares $66.67 (= 0.667 $100) $33.33 (= 0.667 $50) $133.33 (= 0.667 $200) Borrow $30.3 $30.3 $33.33 (= $30.3 1.1) $33.33 (= $30.3 1.1) $36.36 0 $100 (e) No. The true probability of a price rise is almost certainly higher than the risk neutral probability, but it does not help to value the option. 5