JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Date: 31/10/2017 Exercise Session #5 Suggested Solutions Problem 1. (9.21) A project has the following forecasted cash flows: The estimated project beta is 1.5. The market return r m is 16%, and the risk free rate r f is 7%. (a) Estimate the opportunity cost of capital and the project s PV (using the same rate to discount each cash flow). (b) What are the certainty equivalent cash flows in each year? (c) What is the ratio of the certainty equivalent cash flow to the expected cash flow in each year? (d) Explain why this ratio declines. (a) Using the Security Market Line, we find the cost of capital: Therefore: r = 0.07 + 1.5 (0.16 0.07) = 0.205 or 20.5%. P V = 100, 000 + 40, 000 60, 000 50, 000 + + = $3, 090. 2 3 1.07 (b) CEQ 1 = 40, 000 = $35, 520, ( ) 2 1.07 CEQ 2 = 60, 000 = $47, 310, ( ) 3 1.07 CEQ 3 = 50, 000 = $35, 010. 35, 520 (c) a 1 = 40, 000 = 0.888, 47, 310 a 2 = 60, 000 = 0.7885, 35, 010 a 3 = 50, 000 = 0.7002. (d) Using a constant risk adjusted discount rate is equivalent to assuming that a t decreases at a constant compounded rate. 1
Problem 2. (9.22) The McGregor Whisky Company is proposing to market diet scotch. The product will first be test marketed for two years in southern California at an initial cost of $500,000. This test launch is not expected to produce any profits but should reveal consumer preferences. There is a 60% chance that demand will be satisfactory. In this case McGregor will spend $5 million to launch the scotch nationwide and will receive an expected annual profit of $700,000 in perpetuity. If demand is not satisfactory, diet scotch will be withdrawn. Once consumer preferences are known, the product will be subject to an average degree of risk, and, therefore, McGregor requires a return of 12% on its investment. However, the initial test market phase is viewed as much riskier, and McGregor demands a return of 20% on this initial expenditure. What is the NPV of the diet scotch project? At t = 2, there are two possible values for the project s NPV: NP V 2 (if test is not successful) = 0, Therefore, at t = 0: NP V 2 (if test is successful) = $5, 000, 000 + $700, 000 0.12 = $833, 333. NP V 0 = $500, 000 + 0.4 $0 + 0.6 $833, 333 1.2 2 = $152, 778. Problem 3. (20.16) In January 2009, a one year call on the stock of Amazon.com, with an exercise price of $45, sold for $19.55. The stock price was $55. The risk free interest rate was 2.5%. How much would you be willing to pay for a put on Amazon stock with the same maturity and exercise price? Assume that the Amazon options are European options. (Hint: Use put call parity.) Let P = the value of a put, C = the value of a 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 + EX 1 + r = P + S. P = C + EX $45 S = $19.55 + $55 = $8.45. 1 + r 1.025 Problem 4. (20.24) Option traders often refer to straddles and butterflies. Here is an example of each: Straddle: Buy call with exercise price of $100 and simultaneously buy put with exercise price of $100. 2
Butterfly: Simultaneously buy one call with exercise price of $100, sell two calls with exercise price of $110, and buy one call with exercise price of $120. Draw position diagrams for the straddle and butterfly, showing the payoffs from the investor s net position. Each strategy is a bet on variability. Explain briefly the nature of each bet. Straddle: Butterfly: The buyer of the straddle profits if the stock price moves substantially in either direction; hence, the straddle is a bet on high variability. The buyer of the butterfly profits if the stock price doesn t move very much, and hence, this is a bet on low variability. Problem 5. (20.27) The table below lists some prices of options on common stocks (prices are quoted to the nearest dollar). The interest rate is 10% a year. Can you spot any mispricing? What would you do to take advantage of it? Consider each company in turn, making use of the put call parity relationship: V alue of call + P resent value of exercise price = V alue of put + Share price. 3
Drongo Corp. Here, the left hand side $52 + $50 /1.05 = $99.62 is less than the right hand side $20 + $80 = $100. Therefore, there is a slight mispricing. To take advantage of this situation, one should buy the call, invest $47.62 at the risk free rate, sell the put, and sell the stock short. Ragwort, Inc. Here, the left hand side $15 + $100 /1.05 = $110.24 is greater than the right hand side $10 + $80 = $90. Therefore, there is a significant mispricing. To take advantage of this situation, one should sell the call, borrow $95.24 at the risk free rate, buy the put, and buy the stock. Wombat Corp. For the three month option, the left hand side $18+ $40 /1.025 = $57.02 and the right hand side $7 + $50 = $57 are essentially equal, so there is no mispricing. For the first six month option, the left hand side $17 + $40 /1.05 = $55.1 is slightly greater than the right hand side $5 + $50 = $55, so there is a slight mispricing. For the second six month option, the left hand side $10 + $50 /1.05 = $57.62 is slightly less than the right hand side $8 + $50 = $58, and so there is a slight mispricing. Problem 6. (21.1) The stock price of Heavy Metal (HM) changes only once a month: either it goes up by 20% or it falls by 16.7%. Its price now is $40. The interest rate is 12.7% per year, or about 1% per month. (a) What is the value of a one month call option with an exercise price of $40? (b) What is the option delta? (c) Show how the payoffs of this call option can be replicated by buying HM s stock and borrowing. (d) What is the value of a two month call option with an exercise price of $40? (e) What is the option delta of the two month call over the first one month period? (a) Using risk neutral method, V alue of call = p 20 + (1 p) ( 16.7) = 1 p = 0.48. 0.48 (1.2 $40 $40) + (1 0.48) $0 = $3.8. spread of option prices (b) Delta = spread of stock prices = 0.2 40 0 1.2 40 (1 0.167) 40 = 8 14.68 = 0.545. (c) The replicating portfolio is the following: Current Possible Future Cash Flow Cash Flows Buy call $3.8 0 $8 equals Buy 0.545 shares $21.8 (= 0.545 $40) $18.2 (= 0.545 (1 0.167) $40) $26.2 (= 0.545 1.2 $40) Borrow $18 $18 $18.2 (= $18 ) $18.2 (= $18 ) $3.8 0 $8 (d) Possible stock prices with call option prices in parentheses: 4
Option prices were calculated as follows: 0.48 0 + 0.52 0 0.48 17.6 + 0.52 0 Month 1: = 0 and 0.48 8.4 + 0.52 0 Month 0: = $4. spread of option prices (e) Delta = spread of stock prices = 8.4 0 48 33.32 = 0.572. = $8.4. Problem 7. (21.10) Suppose a stock price can go up by 15% or down by 13% over the next year. You own a one year put on the stock. The interest rate is 10%, and the current stock price is $60. (a) What exercise price leaves you indifferent between holding the put or exercising it now? (b) How does this break even exercise price change if the interest rate is increased? (a) Let p equal the probability of a rise in the stock price. Then, if investors are risk neutral: p 15 + (1 p) ( 13) = 10 p = 0.821. The possible stock prices next period are: $60 1.15 = $69, $60 0.87 = $52.2. Let X equal the break even exercise price. Then the following must be true: X 60 = p $0 + (1 p) (X $52.2). 1.1 That is, the value of the put if exercised immediately equals the value of the put if it is held to next period. Solving for X, we find that the break even exercise price is $61.52. (b) If the interest rate is increased, the value of the put option decreases. 5