This Document Contains Chapters 25 to 26 Solutions to Chapter 25 Options 1. Profit and payoff if final stock price = $84 Cost Payoff Profit a) Put option, X = 80 3.95 0 - 3.95 b) Call option, X = 80 7.95 4 - 3.95 c) Put option, X = 84 5.75 0 - 5.75 d) Call option, X = 84 5.50 0 - 5.50 e) Put option, X = 88 7.75 4 - 3.75 f) Call option, X = 88 3.55 0 - 3.55 2. a. Profit and payoff if final stock price = $90 Cost Payoff Profit a) Put option, X = 80 3.95 0 - 3.95 b) Call option, X = 80 7.95 10 + 2.05 c) Put option, X = 84 5.75 0 - 5.75 d) Call option, X = 84 5.50 6 + 0.50 e) Put option, X = 88 7.75 0 - 7.75 f) Call option, X = 88 3.55 2 - 1.55 b. Profit and payoff if final stock price = $78 Cost Payoff Profit a) Put option, X = 80 3.95 2 - 1.95 b) Call option, X = 80 7.95 0 - 7.95 c) Put option, X = 84 5.75 6 + 0.25 d) Call option, X = 84 5.50 0 - 5.50 e) Put option, X = 88 7.75 10 + 2.25 f) Call option, X = 88 3.55 0 - 3.55 3. a. The July call costs $1.27. The January call costs $3.55. b. The January call costs more because there is more uncertainty today about the stock price in January than in July, and uncertainty surrounding the stock price makes options more valuable. With the longer time to expiry, there is more opportunity for the stock price to change. 25-1 This Document Contains Chapters 25 to 26 c. This is true of puts as well. The July put with exercise price $88 sells for $5.15, while the January put sells for $7.75. Likewise, the July put with a $84 exercise price sells for $2.86, while the January put sells for $5.75. 4. Call; exercise; put; exercise. 5. Figure 25.7a represents a call seller; Figure 25.7b represents a call buyer. Figure for put buyer: Figure for put seller: 6. Consider options with exercise prices of $100. Call the stock price at maturity S. Payoff to option position at expiration S < 100 S ≥ 100 Call option buyer 0 S – 100 Put option seller S – 100 0 Stock price Value of investment at maturity Stock price Value of investment at maturity 25-2 While both call buyers and put sellers hope that the stock price increases, the positions are not equivalent. The figure for the call seller is Figure 25.7a. The figure for the put seller is shown in the answer to question 5, above. 7. Payoff to option position at expiration S < 100 S ≥ 100 Put option 100 - S 0 Stock S S Total payoff 100 S a. As seen above, if the stock price, S, is below the exercise price of the put option, $100, your payoff is $100. (i.e., you would exercise the option to sell the stock for the exercise price). b. If the stock price is above $100, the payoff is the value of the stock (i.e., you would throw away the put and either keep the stock or sell it at the market price). The figure below shows the payoff if stock price is $50 (payoff = $100), and payoff if stock price is $200 (payoff =$200) 8. Consider options with exercise price of $100. Call the stock price at maturity S. a. Value of asset at option expiration S < 100 S ≥100 Buy a call 0 S – 100 Invest PV(100) 100 100 Total 100 S 200 Stock price Value of investment at maturity 100 100 payoff from put payoff from stock payoff from put and stock 200 50 25-3 b. Value of asset at option expiration S < 100 S ≥ 100 Buy a share S S Buy a put 100 – S 0 Total 100 S Figure is the same as the figure for problem 7. c. Value of asset at option expiration S < 100 S ≥100 Buy a share S S Buy a put 100 – S 0 Sell a call 0 100 – S Total 100 100 Figure is on next page. E E stock price payoff from bank deposit payoff from call payoff from call and bank deposit Value at maturity 25-4 d. Value of asset at option expiration S < 100 S ≥ 100 Buy a call 0 S – 100 Buy a put 100 – S 0 Total 100 – S S – 100 Stock price Value of investment at maturity 100 100 payoff from put payoff from investment Stock price Value of investment at maturity E E payoff from put payoff from stock payoff from selling call payoff from call payoff from put payoff from put and call stock price Value at maturity 25-5 9. a. To create this pay off buy a call and a put with the same exercise price and same maturity at the same time. The package of investments that would provide the set of payoffs depicted in Figure 25.8 is buy an October 2011 call with exercise price $84 and buy an October 2011 put with exercise price $84. b. The March 2011 price of the put and call with $84 exercise price that mature in October 2011 are these: $4.60 + $4.45 = $9.05 c. When the investor anticipates that there will be big swings of stock price. Table 25.2 indicates that, in March 2011, the price of TD stock was $84.45. An investor who believed that the price of TD would vary considerably between March 2011 and October 2011 might invest in this package. The greater the change in price from $84.45, the greater the payoff to the straddle position. 10. The lower bound is the option’s value if it expired immediately: either zero, or the stock price less the exercise price, whichever is greater. The upper bound is the stock price. 11. a. Zero b. Stock price less the present value of the exercise price. 12. Let X denote the exercise price. Payoff to option position at expiration S<80 80 <S <84 84< S <88 S ≥ 88 Buy call (X = 80) 0 S – 80 S – 80 S – 80 Sell 2 calls (X = 84 ) 0 0 −2(S − 84) −2(S − 84) Buy call (X = 88) 0 0 0 S – 88 Total 0 S – 80 88 – S 0 Notice that, if S is between 84 and 88 at option maturity, the total payoff to the portfolio is positive. Otherwise it is zero. This portfolio has only nonnegative payoffs. But it will cost zero to purchase if the X = 84 option costs the average of the options with X = 80 and X = 88. This situation cannot persist. At a cost of zero, all investors will attempt to buy an unlimited amount of this portfolio. We conclude that the cost of the X = 84 option must be less than the average of the other two options. In this case, the cost of establishing this portfolio is positive, which is consistent with the nonnegative payoffs to the portfolio. 13. a. Decreases b. Increases 25-6 c. Decreases (The present value of the exercise price decreases.) d. Increases e. Decreases f. Decreases 14. Internet: Canadian Stock Options Options quotes are found at http://www.m-x.ca/nego_cotes_en.php, then search using “Equity Options”. Select a stock ticker symbol and find the option prices. Expected results: Students should be able to find examples to illustrate the relationships between option prices and exercise prices and time to expiry. TD – Toronto-Dominion Bank (The) Last update: Sept. 10, 2011, 13:55 Montréal time - (DATA 15 MINUTES DELAYED) Refresh | Print Last Price: 74.220 Net Change: -2.090 Bid Price: 74.170 Ask Price: 74.240 30-Day Historical Volatility: 33.14% Calls Month / Strike Bid Price Ask Price Last Price Impl. Vol. Vol. + 11 SE 60.000 14.150 14.350 16.400 78.74 0 + 11 SE 70.000 4.400 4.550 4.750 39.64 10 + 11 SE 80.000 0.000 0.070 0.070 N/Av 100 + 11 SE 90.000 0.000 0.060 0.040 N/Av 0 + 11 OC 60.000 14.350 14.550 16.500 47.33 0 + 11 OC 70.000 5.350 5.550 6.750 31.43 2 + 11 OC 80.000 0.470 0.580 0.560 23.43 11 + 11 OC 90.000 0.000 0.090 0.110 N/Av 30 Puts Month / Strike Bid Price Ask Price Last Price Impl. Vol. Vol. + 11 SE 60.000 0.020 0.070 0.060 73.19 0 + 11 SE 70.000 0.240 0.250 0.230 37.37 130 + 11 SE 80.000 5.750 5.900 3.850 N/Av 0 + 11 SE 90.000 15.700 15.900 13.800 N/Av 0 + 11 OC 60.000 0.410 0.460 0.480 46.13 10 + 11 OC 70.000 1.590 1.650 1.590 33.19 43 + 11 OC 80.000 6.800 7.000 6.700 28.65 3 25-7 + 11 OC 90.000 16.350 16.550 14.400 42.07 0 The higher the strike price, the lower the call option price; the higher the strike price, the higher the put option price. The longer time to maturity the higher the price is for both calls and put options. 15. Click on the “Options List” to see the list of various options traded on the Montreal exchange. In addition to options on individual stocks, options based on the TSX stock indexes, on ETFs (exchange traded funds) and US dollar are traded on the Montreal Exchange. They are structured like an option on a stock, with exercise prices and expiration dates. The only difference is the underlying security. For example, the SXO option is an option on the S&P/TSX 60 Index. This index is a value-weighted index of the 60 large companies listed on the TSX, selected to reflect the balance of industries listed on the exchange. If you buy the index, you are investing in the equity of 60 different companies. A call option on the index gives the holder the right to buy the underlying index at the exercise price, on or before the option expiration. 16. If you hold call options you will be more tempted to choose the high-risk proposal. This increases the expected payoff and the value of your options. (Notice that the options introduce a potential conflict of interest between managers and shareholders. The options can lead managers to prefer projects with high volatility.) 17. a. call ( it is a real option) b. put (it is a financial option) c. put (You can think of the ability to sell the machine as analogous to a put option with “exercise price” equal to the price of the machine in the secondhand market. Another real option) 18. a. The project gives the firm a call option to pursue (buy) a project [the project is potential expansion of the existing switching project] in the future if that project turns out to be valuable. b. The real option is the ability to sell the equipment. This is a put option which would not be available if the firm were to choose the more specialized production line. 19. a. This is a 5-year call option on gold. The exercise price is $500 an ounce, the price at which the gold can be extracted. b. This is a put option to abandon the restaurant for an exercise price of $5 million. The restaurant’s current value is given by the perpetuity value of the 25-8 annual $700,000 per year. This would be 700,000/(required rate of return,r) which is the value of the “underlying asset.” 20. The price support system gives farmers the right to sell their crop to the government. This is a put option with exercise price equal to the support price. 21. a. If the portfolio value exceeds the threshold, the manager’s bonus is proportional to the difference between the portfolio value and the threshold. If the stock price is below the threshold, the manager gets nothing. This is just like the payoff to a call option on the portfolio with exercise price equal to the threshold. The call would also provide a payment only if the portfolio value exceeds a threshold value equal to the exercise price. b. Such contracts could induce managers to increase portfolio risk since that would increase the value of their implicit call option. 22. a. Rank and File has an option to put (sell) the stock to the underwriter. If the company fails to sell all shares needed to raise $50 million, the put option allows the company to receive a payment equal to the number of unsold shares times the $25 per share price. The cost of buying the put is $2 million. b. The value of the option depends on the volatility of the stock value, the length of the period for which the underwriter guarantees the issue, the interest rate, the price at which the underwriter is obligated to buy the stock, and the market value of the stock in the absence of the underwriter’s guarantee. 23. a. Because the depositor receives a zero rate of return if the market declines and a proportion of any rise in the market index, the depositor has a payoff that is effectively the same as that on a call option on the index. One way for the bank to hedge its position is to purchase call options on the market index. That way, the options implicitly sold are hedged by the options purchased. b. In this case, the depositor has in effect purchased a put option from the bank. To hedge, the bank should purchase puts to offset the exposure of the puts implicitly sold. 24. The CDIC pays out an amount equal to deposits minus bank assets if assets are insufficient to cover all deposits. This is the payoff to a put option on the bank assets with exercise price equal to the deposits owed to bank customers. 25. The projects would give the U.S. the ability to “buy” energy for the fixed cost of synthetic fuels. This is a call option with exercise price equal to the cost of synthetic 25-9 fuel. Greater uncertainty in oil prices would increase the value of the option, and should make investors willing to spend more to develop such technologies. 26. a. Buy a call option for $3. Exercise the call to purchase stock. Pay the $20 exercise price. Sell the share for $25. Riskless profit equals $25 – ($3 + $20) = $2. As investors rush to pursue this strategy, they will drive up the call price until the profit opportunity disappears. The minimum price of the call option is the greater of 0 or the stock price minus the exercise price: Minimum = 25 – 20 = $5 b. Buy a share and a put option for a total outlay of $25 + 4 = $29. Immediately exercise the put to sell the share for $30. Riskless profit equals $1. The minimum value of a put is the greater of 0 or the exercise price minus the stock price: Minimum = 30 – 25 = $5. 27. At an 8 percent yield to maturity, a 10-year, 6 percent coupon AAA-rated bond ordinarily would sell for $60 × annuity factor (8%, 10 years) + $1000/(1.08)10 = $865.80. Therefore, the conversion option (the implicit call option) must be worth $1050 – $865.80 = $184.20. If the bond were converted into shares, the investor would receive 20 shares. At $50/share, the total share value would be 20 × $50 = $1000. Thus bond is worth more than the shares it can be converted into. This is because the bond has a “floor” value equal to its value as a “straight” 6% coupon bond. Even if the stock price declines significantly, the bond will not sell for less than the straight bond value. This extra protection makes the bond worth more than the shares it can be converted into. 28. This question does not have an answer per se. These website is provided to allow the interested student to learn more about real options. Much of the content, however, requires a deeper knowledge of options than can reasonably be covered in an introductory course. 29. a. The package of investments that would provide the set of payoffs depicted in Figure 25.9 is: buy one October 2011 call with exercise price $80, sell two October 2011 calls with exercise price $84, and buy one October 2011 call with exercise price $88. (Payoffs for a butterfly spread are also shown in the table for the solution to Problem 12.) b. Using the data in Table 25.2 the cost of buying this package in March 2011 is: Sell 2 84 calls +$9.20 25-10 Buy 1 80 call -$7.10 Buy 1 88 call -$2.25 Total -$0.15 c. Traders often use the butterfly strategy when they feel a particular stock will remain neutral during a certain period. 30. a. The call will be worth $94 – $84 = $10 if the stock price is $94. It will be worthless if the stock price is $76. b. Payoff at option expiration S = $76 S = $94 9 calls (X = 84) 0 9 ×$10 = $90 Value of 5 shares 5×$76 = $380 5 × $94 = $470 Repay loan −$380 −$380 Total 0 $180 c. The net outlay required to buy the 5 shares and borrow the PV of $380 is 5 × $84 – $380/1.0116 = 420 –375.64 = $44.36 d. The 9 calls also must sell for $44.36, since the payoff to the calls is identical to the payoff to the share-plus-borrowing position. Therefore, each call option must sell for $44.36/9 = $4.93 e. The option in this problem is worth more than the value derived in the example in the chapter. This is because the higher volatility of the stock price assumed in this example. Here, the range of the possible stock prices in September is $94 - $76, or $18. In the example, the range was only $14 (= $92 - $78). Higher stock price volatility increases the value of the call. 31. Return to the example in the chapter. If the interest rate were zero, then your promise to pay $380 at option maturity would bring in $380 today. The net outlay for Strategy B (buy 5 shares and borrow PV of $380) would then be 5 × $84 – $380 = $40. Each option would be worth $40/10 = $4. The call was worth more ($0.93) when the interest rate was higher, 2%. The call option value falls when interest rates fall and the call option value rises when interest rates rise. 32. Internet: Gathering data to implement the Black-Scholes Option Pricing Model. Expected results: Adventurous students will have the thrill of pricing options with data they have collected. 25-11 Treasury Bills - 1 year GRAPH PERIOD: 8 September 2010 - 7 September 2011 ECA – EnCana Corporation Last update: Sept. 10, 2011, 14:55 Montréal time - (DATA 15 MINUTES DELAYED) Last Price: 22.810 Div Yield: 0.19 Net Change: -0.840 Bid Price: 22.770 Ask Price: 22.810 30-Day Historical Volatility: 28.40% Calls Month / Strike Bid Price Ask Price Last Price Impl. Vol. Vol + 12 JA 24.000 1.310 1.450 1.390 35.76 34 + 12 JA 25.000 0.970 1.120 1.070 35.37 20 Input Values Options Price Value: $0.828 Share Price: 22.81 Strike Price: 24.00 Maturity(yrs): 0.25 Dividend Yield: 0.19 Interest Rate: 0.9 Volatility: 28.4 Input Values Options Price Value: $0.54 Share Price: 22.81 Strike Price: 25.00 Maturity(yrs): 0.25 Dividend Yield: 0.19 Interest Rate: 0.9 Volatility: 28.4 ABX – Barrick Gold Corporation Last update: Sept. 10, 2011, 15:13 Montréal time - (DATA 15 MINUTES DELAYED) Refresh | Print Last Price: 54.390 Div Yield: 0.12 Net Change: -0.220 Bid Price: 54.390 Ask Price: 54.410 30-Day Historical Volatility: 33.60% Calls Month / Strike Bid Price Ask Price Last Price Impl. Vol. Vol. + 12 JA 56.000 3.550 3.700 3.600 32.99 59 + 12 JA 58.000 2.820 2.970 2.920 33.05 97 Date Yield 2011-09-07 0.90 25-12 Input Values Options Price Value: $2.988 Share Price: 54.39 Strike Price: 56 Maturity(yrs): 0.25 Dividend Yield: 0.12 Interest Rate: 0.9 Volatility: 33.60 Input Values Options Price Value: $2.264 Share Price: 54.39 Strike Price: 58 Maturity(yrs): 0.25 Dividend Yield: 0.12 Interest Rate: 0.9 Volatility: 33.60 The calculated call prices are results from www.numa.com option calculator. The calculated prices are different from the c-x list prices. When standard deviation is changed, the call price will change accordingly, the higher the standard deviation, the higher the call price. 25-13 Brealey 5CE Solutions to Chapter 26 1. Insurance is a part of risk management that is similar in many ways to hedging. Both activities are designed to eliminate the firm’s exposure to a particular source of risk. Hedging and insurance have several advantages. They can reduce the probability of encountering financial distress, or in extreme cases, bankruptcy. They make the firm’s performance less vulnerable to events not in the firm’s control, and therefore enable managers and investors to better evaluate performance. By reducing the impact of such random events, they also facilitate planning. However, insurance is costly. Furthermore, by holding a diversified investment portfolio, shareholders can costlessly eliminate firm-specific risks (Look back at Chapter 10). Why should a company spend a lot of money on insurance to reduce the impact of risks of no concern to shareholders? Think about: should managers buy enough insurance to reduce risk to zero? NO! This would be far too costly and would lower the expected return on the company’s equity. Hedging and insurance make most sense when the source of uncertainty has a significant impact on the firm’s performance. This is especially true if the risk will result in costly financial distress or bankruptcy. Hedging and insurance can be value-increasing if the company avoids encountering the costs of financial distress and bankruptcy. (See Chapter 15). It is not worth the time or effort to protect against events that cannot materially affect the firm. 2. a. She should sell futures. If interest rates rise and bond prices fall, the gain on the futures will offset the loss on the bonds. b. He should sell futures. If interest rates rise and bond prices fall, the profits on the futures will offset the lower price the firm will receive for its bonds. 3. Scan The Globe and Mail’s Report on Business and The National Post (including their online editions) and you will see a wide array of information on websites from which you can learn more about futures on agricultural commodities, as well as other raw materials. Commodities futures traded on ICE Futures Canada include wheat, flax and barley. Other commodities, including gold, silver, tin, gas, sugar, cotton and corn futures are traded on futures exchanges around the world. Purchasers, including manufactures and food processors, of any of these commodities can hedge cost risk by buying futures contracts, and producers can hedge revenue risk by selling futures contracts. 26-1 4. The object of hedging is to eliminate risk. If you eliminate risk, you will eliminate the happy surprises as well as the unhappy ones. If the farmer wishes to lock in the value of the canola, then it is inconsistent to argue later that he is subject to the risk of losing the opportunity to sell canola at a price higher than $580. Farmers who are in a position to benefit from increases in canola prices are at least implicitly speculating on canola prices, not hedging. 5. The contract size is 5000 bushels, so the farmer who sells a wheat futures contract realizes the following cash flows on each contract: Cash flow per contract Futures Change in (5000 × decrease price futures price in futures price) Day 1 $3.83 0 0 Day 2 $3.98 +.15 − 750 Day 3 $3.70 −.28 +1,400 Day 4 $3.50 −.20 +1,000 Day 5 $3.60 +.10 − 500 Total: −.23 +1,150 The sum of the mark to market cash flows equals the total decrease in the futures price times 5000: -(3.60 – 3.83) × 500 5000= 1,150. This is the same payment that would be required if the contract were not marked to market. Only the timing of the cash flows is affected by marking to market. 6. One advantage of holding futures is the greater liquidity (ease of purchasing and selling positions) than is typical in the spot market for a commodity. Another is the fact that you do not need to buy the commodity. You simply put up the margin on the contract until the maturity date of the contract. This saves you the opportunity cost of capital on those funds for the length of the contract. A third advantage is that, if you hold futures instead of the underlying commodity, you will not incur storage, insurance, or spoilage costs on the commodity. The disadvantage of the futures position is that you do not receive the benefits that might accrue from holding the underlying asset. For example, holding stock rather than stock index futures allows the investor to receive the dividends on the stocks. Holding copper inventories rather than copper futures allows a producer to avoid the types of inventory shortage costs we discussed in Chapter 20. 26-2 7. Gold price $1,330 $1,450 $1,650 a. Revenues $1,330,000 $1,450,000 $1,650,000 Futures contract 130,000 10,000 -190,000 b. Total $1,460,000 $1,460,000 $1,460,000 c. Revenues 1,330,000 $1,450,000 $1,650,000 +Put option payoff 120,000 0 0 −Put option cost 5,000 5,000 5,000 Net revenue $1,445,000 $1,445,000 $1,645,000 8. Gold price $1,330 $1,450 $1,650 a. Cost of gold $1,330,000 $1,450,000 $1,650,000 +Cost of call 3,000 3,000 3,000 −Call option payoff 0 0 200,000 Net outlay $1,333,000 $1,453,000 $1,453,000 b. Cost of gold $1,330,000 $1,450,000 $1,650,000 +Cost of call 7,000 7,000 7,000 −Call option payoff 0 5,000 205,000 Net outlay $1,337,000 $1,452,000 $1,452,000 The more expensive but lower exercise price calls give you a better outcome (a lower net outlay) when the price of gold is high, but a worse outcome when the price of gold is low. 9. Suppose you lend $100 today, for one year, at 6% and borrow $100 today for two years at 7%. Your net cash flow today is zero. In one year, you will receive $106, and you will owe $100 × (1.07)2 = $114.49 for payment one more year hence. This is effectively a one-year borrowing agreement at rate 114.49/106 – 1 = .08, which is the forward rate for year 2. Since you can create a "synthetic loan" at the forward rate, which is less than the bank’s offer, you should reject the offer. 10. One way to protect the position is to sell 10 million yen forward. This locks in the dollar value of the yen you will receive if you get the contract. However, if you do not receive the contract, you will have inadvertently ended up speculating against the yen. Suppose the forward price for delivery in 3 months is ¥105/$, and you agree to sell forward 10 million yen or $95,238. 26-3 ¥100/$ ¥110/$ If you win the contract: Dollar value of contract $100,000 $ 90,909 Profit on forward contract − 4,762 4,329 Total $ 95,238 $95,238 If you don’t win the contract: Dollar value of contract $ 0 $ 0 Profit on forward contract − 4,762 4,329 Total − $ 4,762 $ 4,329 If you win the contract, the forward contract locks in the dollar value of the contract at the forward exchange rate. However, if you lose the contract, then your short position in yen will result in losses if the yen appreciates and gains if it depreciates. This is a speculative position. Another approach would be to buy put options on yen. If you buy options to sell 10 million yen at an exercise price of ¥105/$, then if you win the contract, you are guaranteed an exchange rate no worse than ¥105/$ if the yen depreciates, but you can benefit from appreciation in the yen. This appears to be superior to the forward hedge, but remember that the options hedge requires you to purchase the put. This hedge can be costly. 11. Petrochemical will take a long position to hedge its cost of buying oil. Onnex will take a short position to hedge its revenue from selling oil. Cost for Petrochemical Oil price ($ per barrel) $88 $90 $92 Cost of 1000 barrels −$88,000 −$90,000 −$92,000 + Cash flow on long futures position − 2,000 0 2,000 Net cost −$90,000 −$90,000 −$90,000 Revenue for Onnex Oil price ($ per barrel) $88 $90 $92 Revenue from 1000 barrels $88,000 $90,000 $92,000 + Cash flow on short futures position 2,000 0 − 2,000 Net revenues $90,000 $90,000 $90,000 26-4 The benefit of futures is the ability to lock in a riskless position without paying any money. The benefit of the option hedge is that you benefit if prices move in one direction without losing if they move in the other direction. However, this asymmetry comes at a price: the cost of the option. 12. You would receive the gold at the maturity date, and you would pay the futures price on that date. Your total payments, including the net proceeds from marking to market, equal the futures price on the day that you enter the futures contract. The futures price is greater than the spot price for gold. This reflects the fact that the futures contract ensures your receipt of the gold without tying up your money now. The difference between the spot price and the futures price reflects compensation for the time value of money. Another way to put it is that the spot price must be lower than the futures price to compensate investors who buy and store gold for the opportunity cost of their funds until the futures maturity date. 13. The car manufacturers could have bought dollars forward for a specified number of deutschemarks (or equivalently, sold deutschemark contracts). This would serve to hedge total profits because, when profits in the U.S. market decrease due to appreciation in the deutschemark, the company would realize greater profits on its futures or forward contracts. On the other hand, it is less clear that such hedging would improve the competitive position of the manufacturers. Once the contracts are in place, each firm should still evaluate the car sale as an incremental transaction that is independent of any proceeds from the forward position. Is the dollar price charged on each incremental sale enough to cover the incremental costs incurred in deutschemarks? If not, the firm may decide to raise the dollar price regardless of any profits on futures contracts. So, while the hedge can stabilize the value of overall profits in the face of currency risk, it is less likely to affect the competitive position of the firm. 14. A currency swap is an agreement to exchange a series of payments in one currency for a given series of payments in another currency. For example, a Canadian firm that will be buying £1 million of supplies each year from a British producer might enter into a currency swap in which it pays $1.50 million a year and receives £1 million pounds in return. This arrangement locks in the dollar cost of the parts purchased from the U.K. supplier. 26-5 An interest rate swap is an exchange of a series of fixed payments for a series of payments linked to market interest rates. For example, a bank that pays its depositors an interest rate that rises and falls with the level of general market rates might enter a swap to exchange a fixed payment of $60,000 a year for a floating payment equal to the T-bill rate times $1 million. (For example, if the T-bill rate is 4 percent, then the floating rate payment would be $40,000.) This arrangement locks in the amount of the bank’s expenses: the receipt of floating rate payments offsets the payments to depositors, so the bank is left with only the fixed payments on the swap agreement. 15. Notice that while Firm A pays a higher interest rate in both markets (presumably because it presents greater default risk), it has a relative disadvantage in borrowing in Canada, where its cost of funds is 2% higher than Firm B’s. In comparison, it gets relatively better terms in euro countries, where its cost of funds is only 1% higher than Firm B’s. Instead of borrowing in the desired currency, each firm should borrow in the currency in which it has a comparative advantage. Then the firms can swap cash flows back into the desired currency. Suppose that A sets a goal of reducing its dollar interest rate to 9.5%, which is .5% lower than its rate on a dollar loan. Any additional savings from the swap will accrue to Firm B. Here is how the swap might work. Step 1: B borrows $1000 at an 8% rate in Canada (where it has a comparative advantage) and is obligated to pay $80 a year for 4 years and $1080 in the fifth year. Step 2: Firm A calculates that the present value of Firm B’s debt payments using its own target 9.5% interest rate is $942.40. Therefore, Firm A borrows this amount of money in a European country, such as Switzerland, where it has its comparative advantage. It borrows 942.40 euros. At a 7% interest rate, Firm A must pay .07 × 942.40 = 65.97 euros each year for 4 years and then pay 65.97 + 942.40 = 1008.37 euros in the fifth year. Step 3: The two firms enter a swap arrangement to exchange cash flows equal to the other’s principal and interest payments. So Firm A pays Firm B $80 and receives 65.97 euros each year for four years. In year 5, Firm A pays $1080 in return for 1008.37 euros. Firm A’s net cash flows are as follows: It initially receives $942.40 by exchanging the proceeds from its euro borrowing into dollars. It uses the income from the swap to pay its euro bonds, and pays $80 a year for 4 years and $1080 in year 5 on the swap. The effective interest rate (yield to maturity) on this loan is 9.5%, which is better than it could have done in Canada. (To determine the yield, think of Firm A as effectively issuing a five-year 8% coupon bond for a price of $942.40.) 26-6 Firm B receives $1,000 initially, which it exchanges for 1,000 euros. Its net cash outflows in the following years are 65.97 euros per year for 5 years and an additional payment of 942.40 euros in the fifth year. This corresponds to a YTM of 5.57%, which is a better rate than it could have obtained by borrowing in euros directly. (This is the yield to maturity on a bond sold with a coupon rate of 7%, face value 942.40, at a price of 1,000.) Both parties receive a better rate than if they had borrowed directly in their preferred currencies. 16. a) If risk management reduces the likelihood of financial distress and bankruptcy, then the (expected) costs of financial distress and bankruptcy are lowered. Recall from Chapter 15 that the costs of financial distress depend on both the probability of distress and the costs encountered is distress occurs. Therefore, it is true that successful risk management strategy can increase the debt capacity. b) This statement is true. Generally, risks that can be costlessly eliminated by management should be eliminated. However, if management has to spend resources to eliminate risk, they should consider whether such actions will increase shareholders’ wealth. For example, it might be possible to engage in such a wide range of hedging and risk-reducing activities that all risks are eliminated. In this case, the expected return on the stock will be same as a risk- free security! If all risk is eliminated, both happy (positive) and unhappy (negative) outcomes are eliminated. Many investors are willing to accept risk because they can earn a higher expected return. Thus management’s job should not be to eliminate all risks. They are paid to decide which risks are worth taking. The success of their strategy will be reflected in the share price and the stock return earned over time. 17. A case study of mishandling financial risk Expected results: Students see how both individual and institutional decisions can lead to mistakes in risk management. This is a rather long document. 18. Managing a business crisis Expected results: The articles provided on this website change but are all related to managing business risks. Students will get a flavour for how to manage business risk, beyond using financial instruments. As we prepared the solutions manual the following items were available through this web site: http://www.smallbusinessnotes.com/business-finances/risk-management- 26-7 strategies.html http://advertising.about.com/od/crisismanagement/Crisis_Management.htm 19. Canadian futures contracts Expected results: Futures contracts are offered on S&P/TSX market indexes (listed “index derivatives”, contract details at http://www.m- x.ca/produits_indices_sxf_en.php) and on Government of Canada debt (listed under “interest rate derivatives”, contract details for money market instruments at http://www.m-x.ca/produits_taux_int_bax_en.php). Students will think of situations in which an investor wants to lock in an interest rate. 20. Careers in risk management Expected results: A list of skills including technical expertise, willingness to learn, ability to communicate, advanced finance and math training etc. 21. Standard & Poor’s: Risk management Expected results: Students will gain insights into risk management activities of an actual company. 26-8 Appendix 6B: Duration: Measuring the Life of a Bond Bond prices rise when interest rates fall, and prices go down when interest rates rise. This gives rise to interest rate risk. But as Figure 6.8 illustrated, not all bonds are equally affected by changes in interest rates. We found that the 30-year bond’s price was more sensitive to a change in the interest rate than the otherwise identical 3-year bond’s price. We concluded that longer-term bonds have more interest rate risk than shorter-term bonds. Up until now, we have considered the number of years to maturity as the measure of the length of a bond’s life. This measure works well if the bond is a zero-coupon bond. A five-year, zero-coupon bond makes one payment after five years. However, if the bond pays coupons, it makes payments each year for five years. To say it has a five-year life is misleading. The average time to the bond’s cash flow is less than five years. Would you say that a 5-year bond with a 10 percent coupon rate has an equal life to an otherwise identical 5-year bond with only a 5 percent coupon rate? With the 10 percent coupon, you receive greater payments during years 1 to 4 than with the 5 percent coupon. The higher the coupon, the sooner you get payment and the shorter the effective life of the bond. We need a new measure of the bond’s life that recognizes that some payments are made sooner than others. Bond analysts often use the concept of duration to describe the life of a bond.1 duration A measurement of the life of a bond based on the average time to the bond’s payments. Duration is a weighted average of the time to each payment. The weight for the time of each payment equals the present value of the payment as a fraction of the bond’s total value. If P0 is the current value of an n-year bond with annual coupon payments, C1, C2, … ,Cn are the cash flows (coupon payments plus face value) for years 1 to n, and PV(C) is the present value of the cash flows, the bond’s duration is 1 2 3 0 0 0 0 Duration 1 PV( ) 2 PV( ) 3 PV( ) ... PV( ) = × PC + × PC + × PC + + n× PC n (6A.1) Table 6A.1 shows the duration calculation for a 5 percent, 5-year bond paying interest annually and a current price of $957.88. The bond’s yield to maturity is 6 percent. Column 3 shows the present value of each cash flow and column 4 shows its fraction of the bond’s value. For example, the present value of the $50 payment due in one year is $47.17 ($50/1.06). This payment represents 4.9 percent of the bond’s present value, $41.17/$957.88. In other words, at the end of one year, 4.9 percent of the bond’s current value will be received. The present value of the fifth-year payment is $784.62, almost 82 percent of the bond’s value. To calculate a bond’s duration, multiply the number of years to each payment by its proportion of the bond’s value and add up the terms. The duration of the 5-year, 5 percent coupon bond in Table 6A.1 is about 4.5 years. By contrast, the duration of a 5-year, 10 percent coupon bond with a current price of $1,165.80 and a 6 percent yield to maturity is only 4.2 years. As expected, the higher the coupon rate, the shorter the effective maturity of the bond. Table 6B.1 Calculation of duration for a 5 percent, 5-year bond paying interest annually. The bond’s current price is $957.88 and its yield is 6 percent. The bond’s duration is 4.535 years. Present Value of Cash Proportion of Total Present Value of Cash Flow as Proportion of Value Weighted Year Cash flow Flow (in dollars) Total Value Time (in years) 1 50 50/(1.06) = 47.17 47.17/957.88 = 0.049 1 × .049 = 0.049 2 50 50/(1.06)2 = 44.50 45.50/957.88 = 0.046 2 × .046 = 0.093 3 50 50/(1.06)3 = 41.98 41.98/957.88 = 0.044 3 × .044 = 0.131 4 50 50/(1.06)4 = 39.60 39.60/957.88 = 0.041 4 × .041 = 0.165 5 1,050 1,050/(1.06)5 = 784.62 784.62/957.88 = 0.819 5 × .819 = 4.096 Total = 957.88 1.00 4.535 Duration provides a useful way to compare the interest rate risk of bonds. The greater a bond’s duration, the greater the bond’s interest rate risk. Consider again the 5 percent and 10 percent coupon bonds. In Table 6A.2, the prices of each bond are shown as interest rates change from the current 6 percent. For a 50-basis- point increase in the discount rate to 6.5 percent, the 5 percent coupon bond’s price decreases 2.11 percent, but the 10 percent coupon bond’s price decreases only 2.01 percent. Likewise, a 50-basis-point decrease in the discount rate results in a bigger percentage price increase for the 5 percent coupon bond. The results show that the longer a bond’s duration, the more interest rate risk it bears. Table 6B.2 Comparing the interest rate risk of bonds with different coupons but identical maturity. The 5% coupon bond’s duration is greater than the 10% coupon bond’s. The 5% bond’s interest rate sensitivity is also greater than the 10% bond’s. 5-year, 5% coupon (Duration = 4.5 years) 5-year, 10% rate bond (Duration = 4.2 years) Discount rate Bond price Percentage Change from Current Price Bond price Percentage Change from Current Price 5.5% 978.65 +2.17% 1,192.16 +2.03% bond’s 6% (current rate) 957.88 0 1,168.49 0 6.5% 937.66 −2.11% 1,145.45 −2.01% Total percentage 4.28% 4.04% change Check Point 6B.1 Following Table 6A.1, verify that the duration of a 5-year, 10 percent coupon bond with a current price of $1,165.8 and a 6 percent yield to maturity is 4.2 years. Duration is a weighted average of the time to each payment. If the bond pays interest semi-annually the years will need to include half years. For example, a 3-year 4 percent coupon bond paying interest semi- annually, with $1,000 face value, will pay 6 coupon payments of .04/2 x $1000 = $20. The cash flows are paid in .5, 1, 1.5, 2, 2.5, and 3 years. So, first payment on this bond will be $20 half a year from its most recent interest payment and the next payment will be $20 in one year. Calculate the duration of a bond with semi-annual coupon payments using Equation 6B.1 and n will include fraction of years. Use the semi- annual yield to maturity to calculate the present value of each cash flow. Then multiply each ratio of the present value of each cash flow to the current bond price by the amount of time. The correct amount of time is 0.5 year for the first cash flow. So the first term in the duration calculation is 0.5 x PV(C.5)/P0, where PV(C.5) is the present value of the first payment in half of a year discounted at the semi-annual yield to maturity (ytm). So PV(C.5) =C.5/ytm. The next term in the equation is the payment will be in one year, C1, and will be 1 x PV(C1)/P0 and PV(C1)/(ytm)2. What do you think changing to semi-annual interest rate will do the bond’s duration? Duration is the measure of the bond’s life based on the average time of the bond’s payment. So, the same discount rate but paid semi-annually will have a smaller duration. Duration is an important concept used in the management of interest rate risk. Suppose your firm has promised to make pension payments to retired employees. The discounted present value of these pension payments is $4 million. To meet the pension obligations, the company sets aside $4 million now and invests it in government bonds. However, as interest rates change, the present value of the pension liability and also of the bond investment change. To ensure that the value of the bonds in the fund is always sufficient to meet the pension liability, the bonds are selected to create a bond portfolio with the same duration as the pension obligation. This strategy of matching the duration of assets to liabilities is sometimes called portfolio immunization. Questions and Problems 6A.1. Calculating Duration. The following bond ask prices were recorded on June 1, 2010. Assume each bond pays interest annually on June 1. Bond Price ($) Canada 3.5s of 2013 1036.8 Canada 4.25s of 2015 1086.5 a. What is the yield to maturity on each bond? b. What is the duration of each bond? c. Which bond has the greater interest rate risk? 6A.2. Calculating Duration. Using the bonds from problem 6A.1, recalculate the yields to maturity and the duration, assuming that the bonds pay semi-annual interest. How do the values change? 6A.3. Calculating Duration. A star hockey player has negotiated a contract that pays him $5 million in the first year, $7 million in the second year, and $8 million in the third year. If his personal borrowing rate of interest is 8 percent, calculate the duration of the contract. Solution to Check Point 6B.1 The duration for a 10 percent, 5-year bond paying interest annually is 4.237 years, as shown in the table below. The bond’s current price is $1,165.8 and its yield is 6 percent. Present Value of Cash Present Value of Cash Flow as Proportion of Proportion of Total Value weighted Year Cash flow Flow (in dollars) Total Value Time (in years) 1 100 100/(1.06) = 94.34 94.34/1,168.49 = 0.081 1 × .081 = 0.081 2 100 100/(1.06)2 = 89.00 89.00/1,168.49 = 0.076 2 × .076 = 0.152 3 100 100/(1.06)3 = 83.96 83.96/1,168.49 = 0.072 3 × .072 = 0.216 4 100 100/(1.06)4 = 79.21 79.21/1,168.49 = 0.068 4 × .068 = 0.271 5 1,100 1,100/(1.06)5 = 821.98 821.98/1,168.49 = 0.703 5 × .703 = 3.517 Total = 1168.49 1.000 4.237 1 This measure is also known as the Macaulay duration, after its inventor. A good book that includes the topic of duration and bond yields is Investments, 7th Canadian Edition, by Bodie, Kane, Marcus, Perrakis, Ryan (McGraw- Hill Ryerson, 2011). Brealey 5CE Appendix 6B Solutions 6B.1 a. Using calculator to calculate yield to maturity: Canada 3.5% bond maturing in June 2013: 3 years from June 2010, paying interest annually 3.5%, 2-Year bond: PMT = .035 x 1000 = 35, N= 3, FV = 1000, PV = -1,036.8 YTM (I/Y) = 2.2185% Canada 4.25% bond maturing in June 2015: 5 years from June 2010, paying interest annually 4.25%, 5-Year bond: PMT = .0425 x1000 = 42.5, N= 5, FV = 1000, PV = -1,086.6 YTM (I/Y) = 2.3938% b. Duration of Canada 3.5% bond maturing in June 2013 = 2.902 years Year Cash flow Present Value of Cash Flow Present Value of Cash Flow as Proportion of Total Value Proportion of Total Value x Time (years) 1 35 35/(1.022185) = 34.24 = 34.24/1036.8= 0.0330 = 1 x .033 = 0.033 2 35 35/(1.022185)2 = 33.50 = 33.50/1036.8= 0.0323 = 2 x .0323 = 0.065 3 1035 1035/(1.022185)3 = 969.06 = 969.06/1036.8= 0.9347 = 3 x .9347 = 2.804 TOTAL = 1036.80 1.000 2.902 Duration of Canada 4.25% bond maturing in June 2015 = 4.627 years Year Cash flow Present Value of Cash Flow Present Value of Cash Flow as Proportion of Total Value Proportion of Total Value x Time (years) 1 42.5 42.5/(1.023938) = 41.51 = 41.51/1086.5= 0.0382 = 1 x .04 = 0.038 2 42.5 42.5/(1.023938)2 = 40.54 = 40.54/1086.5= 0.0373 = 2 x .0391 = 0.075 3 42.5 42.5/(1.023938)3 = 39.59 = 39.59/1086.5= 0.0364 = 3 x .0382 = 0.109 4 42.5 42.5/(1.023938)4 = 38.66 = 38.66/1086.5= 0.0356 = 4 x .0373 = 0.142 5 1042.5 1042.5/(1.023938)5 = 926.21 = 926.21/1086.5= 0.8525 = 5 x .8933 = 4.262 TOTAL = 1086.50 1.000 4.627 c. The Canada 4.25% bond has the greater duration so has the greater interest rate risk. 6B.2 a. Using calculator to calculate yield to maturity: Canada 3.5% bond maturing in June 2013: 3 years from June 2010, paying interest semi-annually 3.5%, 2-Year bond: PMT = .035/2 x 1000 = 17.5, N= 2x3=6, FV = 1000, PV = -1,036.8 Semi-annual YTM (I/Y) = 1.082% Canada 4.25% bond maturing in June 2015: 5 years from June 2010 4.25%, 5-Year bond: PMT =.0425/2 x1000 = 21.25, N= 2x5=10, FV = 1000, PV = -1,086.6 Semi-annual YTM (I/Y) = 1.2018% b. Duration of Canada 3.5% bond maturing in June 2013 = 2.882 years Year Cash flow Present Value of Cash Flow Present Value of Cash Flow as Proportion of Total Value Proportion of Total Value x Time 0.5 17.5 17.5/(1.01082) = 17.31 = 17.31/1038.6= 0.0167 = .5 x .0167 = 0.008 1 17.5 17.5/(1.01082)2 = 17.13 = 17.13/1038.6= 0.0165 = 1 x .0165 = 0.017 1.5 17.5 17.5/(1.01082)3 = 16.94 = 16.94/1038.6= 0.0163 =1.5 x .0163 = 0.025 2 17.5 17.5/(1.01082)4 = 16.76 = 16.76/1038.6= 0.0162 = 2 x .0162 = 0.032 2.5 17.5 17.5/(1.01082)5 = 16.58 = 16.58/1038.6= 0.0160 = 2.5 x .0160 = 0.040 3 1017.5 1017.5/(1.01082)6 = 953.87 = 953.87/1038.6= 0.9200 = 3 x .092 = 2.760 Total = 1038.60 1.002 2.882 Duration of Canada 4.25% bond maturing in June 2015 = 4.579 years Year Cash flow Present Value of Cash Flow Present Value of Cash Flow as Proportion of Total Value Proportion of Total Value x Time 0.5 21.25 21.25/(1.012018) = 21.00 = 21.00/1086.5= 0.0193 = .5 x .0193 = 0.010 1 21.25 21.25/(1.012018)2 = 20.75 = 20.75/1086.5= 0.0191 = 1 x .0191 = 0.019 1.5 21.25 21.25/(1.012018)3 = 20.50 = 20.50/1086.5= 0.0189 = 1.5 x .0189 = 0.028 2 21.25 21.25/(1.012018)4 = 20.26 = 20.26/1086.5= 0.0186 = 2 x .0186 = 0.037 2.5 21.25 21.25/(1.012018)5 = 20.02 = 20.02/1086.5= 0.0184 = 2.5 x .0184 = 0.046 3 21.25 21.25/(1.012018)6 = 19.78 = 19.78/1086.5= 0.0182 = 3 x .0182 = 0.055 3.5 21.25 21.25/(1.012018)7 = 19.55 = 19.55/1086.5= 0.0180 = 3.5 x .018 = 0.063 4 21.25 21.25/(1.012018)8 = 19.31 = 19.31/1086.5= 0.0178 = 4 x .0178 = 0.071 4.5 21.25 21.25/(1.012018)9 = 19.08 = 19.08/1086.5= 0.0176 = 4.5 x .0176 = 0.079 5 1021.25 1,021.25/(1.012018)10 = 906.25 = 906.25/1086.5= 0.8341 = 5 x .8341 = 4.171 Total = 1086.50 0.166 4.579 Paying interest semi-annually meaning that cash flow is received sooner and the duration is reduced. So the 2.902 years duration of the Canada 3.5% bond maturing in June 2013 paying annual interest is reduced to 2.882 years it interest is paid semi- annually. This is not a big change. The 4.627 years duration of Canada 4.25% bond maturing in June 2015 paying annual interest is reduced to 4.579 years. This is also a big change. 6B.3 The duration of a contract is calculated using the same equation as the duration of the bond. So a term duration of the contract is: = time to payment x PV(payment)/PVof all payments PV(contract payments) =5/(1.08) + 7/(1.08)2 + 8/(1.08)3 = 4.63 + 6.00 + 6.35 = $16.98 million Contract Duration = [1 x 4.63/16.98] + [2 x 6.00/16.98] + [3 x 6.35/16.98] = 2.1 years The contract duration is 2.1 years Solutions for Appendix 25A The Black-Scholes Option Valuation Model 1. a. At expiry, the option will be worth the greater of either 0 or the difference between the stock price and the exercise price. If the stock price is $200 in one year, the option will be worth $200 - $100 = $100. If the stock price is $50, the option is worth 0. Delta = difference in option payoff at ex piry difference in stock price at expiry= 100 - 0 200 - 50 = 100 150 = 2 3 = .667 To replicate the call, you need to buy 2/3 of a stock or, equivalently, to replicate 3 calls you need to buy 2 shares of stock. b., c. Cash Flow Toda y Payoff in One Year if Stock Price is: 50 200 Strategy A Buy 3 calls ??? 0 3×(200-100)=300 Strategy B Buy 2 shares 2 × -100 = -200 2×50 = 100 2×200 = 400 Borrow PV(100) 100 1.1 = 90.91 -100 -100 Total -109.09 0 300 d. The delta value in a. is 2/3. So buying one call is equivalent to buying 2/3 of share and getting a bank loan. Calculate the equivalent bank loan: If the stock price turns out to be $50 in one year the payoff from the stock is 2/3 x 50 and the payoff from the call option is zero. The difference in the payoffs = 2/3 x $50 – 0 = 2/3 x $50. However, if the stock price turns out to be $200 the payoff from the stock is 2/3 x $200 and from the option is $200 - $100 = $100. So the difference in the payoff = 2/3 x $200 - $100 = 2/3 x $50. With a 10% one-year interest rate the value of the bank loan needed at the start of the year = (2/3 x 50) /(1.1) = 30.303 3 calls payoff the same as 2 shares less the loan repayment. Thus the cost of the 3 calls must equal the net cost of buying 2 shares and the loan proceeds, $109.09. One call is $109.09/3, or $36.36. Value of call option = [delta x current stock price] – [bank loan] = 2/3 x 100 – 30.303 = 36.36 25-1 So the price of one call option is $36.36 (which is the same as the answer in b.,c.) 2. a. P = 10 EX = 12 t = 9/12 years = .75 years σ = .3 per year r = (1.01)9 – 1 = .09369, the 9-month interest rate The value of the call option is V = N(d1) × P – N(d2) × PV(EX) PV(EX) = 12 1.09369 = 10.972 Use the EXCEL function NORMSDIST to find the probabilities N(d1) and N(d2): N(d1) = .4102 N(d2) = .3131 Substituting into the Black -Scholes formula for the call price: V = .4102 × 10 - .3131 × 10.972 = .667 The 9-month call option is worth about $.67 b. With only 3 months to expiry, we expect to find that the call price is lower. There is less time for the stock price to increase above the exercise price. Repeating the steps of (a): P = 10 EX = 12 t = 3/12 years = .25 years σ = .3 per year r = (1.01)3 – 1 = .0303, the 3-month interest rate The value of the call option is V = N(d1) × P – N(d2) × PV(EX) d1= L σ × (t)1/2 N[P/PV(EX)] + σ × 2(t)1/2 = L N[10/10.972] .3×(.75)1/2 + .3×(.75)1/2 2 = -.2271 d2 = d1 – σ × (t)1/2 = -.2271 - .3 × (.75)1/2 = -.4869 PV(EX) = 12 1.0303 = 11.647 N(d1) = .1733 N(d2) = .1378 V = .1733 × 10 - .1378 × 11.647 = .128 The 3-month call option is worth about $.13, much less than the 9-month call option, as expected. P = 9.75 EX = 10 t = 3/12 years = .25 years r = .02 25- 3 d1= L σ × (t)1/2 N[P/PV(EX)] + σ × 2(t)1/2 = L N[10/11.647] .3×(.25)1/2 + .3×(.25)1/2 2 = -.9414 d2 = d1 – σ × (t)1/2 = -.9414 - .3 × (.25)1/2 = -1.091 3. This question cannot be solved directly because it is not mathematically possible to solve the Black-Scholes formula for the volatility, σ. However, it can be solved using the trial-and-error method. The most efficient way to do this is to set up a spreadsheet to solve the Black-Scholes formula. If you do this question by hand, it will involve a lot of calculations. Sophisticated algorithms for solving for the volatility have been developed and are used by professionals. Substituting into the Black-Scholes formula for the call price: N(d2). Use the EXCEL function NORMSDIST to find the probabilities N(d1) and The implied volatility is 30%. 4. Stock Price Standard Deviation of Stock Return Exercise Price Time to Maturity (fraction of year) Effective Annual Interest Rate Call Option Price a. $14 40% $15 0.5 0.0609 1.340 $16 40% $15 0.5 0.0609 2.531 If the stock price is higher, the call price will be higher. b. $13 70% $10 0.75 0.0604 4.726 $13 70% $13 0.75 0.0604 3.317 If the exercise price is higher, the call price will be lower. c. $18 55% $25 0.75 0.0604 1.649 $18 55% $25 0.75 0.0672 1.672 If the interest rate is higher, the call price will be higher. d. $23 40% $20 0.5 0.04 4.457 $23 70% $20 0.5 0.04 6.068 If the stock return volatility is higher, the call price will be higher. e. $23 50% $30 0.25 0.04 0.518 $23 50% $30 0.5 0.04 1.311 If the time to maturity is greater, the call price will be higher. 25- 4 Possible values of σ Black-Scholes call price .1 .169 .2 .364 .3 .558 Appendix B: Answers to selected End-of-Chapter Problems Chapter 1 1. real, executive airplanes, brand names, financial, stock, investment, capital budgeting, financing 8. a. financial d. real f. financial g. real 14. The contingency arrangement aligns the interests of the lawyer and the client. More appropriate when client doesn’t know the lawyer’s skill. 16. Such a plan would burden them with a considerable personal risk tied to the fortunes of the firm. 17. Managers who are more securely entrenched in their positions are more able to pursue their own interests. Chapter 2 3. Options markets, foreign exchange markets, futures markets, commodity markets, money market. 4. Buy shares in an exchange-traded fund or a mutual fund. 8. Look up the price of gold in commodity markets, and compare it to $2,500/6 = $416.67/ounce. 19. a. Find the rate of return available on other riskless investments, e.g., 1-year maturity Treasury bills. b. The opportunity cost is 20%, the same expected rate of return available on other investments of comparable risk. The sequester is expected to pay $115,000 on a $100,000 investment, a gain of $15,000. If the $100,000 was invested in the London Carbon Exchange, the expected payback is .2 × $100,000, or $20,000. The purchase of additional sequesters is not a worthwhile capital expenditure. Chapter 5 1. a 46.32 2. a 215.89 4. PV = $548.47 5. 5%, 8%,0% 7. PV = $812.44 8. a. t= 23.36 9. a. 671.01 10.a. 1,448.66 11. 12.68%, 8.24%, 10.25% 12. 9.57%, 6%, 8% 13. n = 9.01 years 15. APR = 52%; EAR = 67.77% 22. a. EAR = 6.78%, APR = 6.57%, compounded monthly, b. PMT = 573.14 23. a. r = 11.11%, b. r = 1/(1 − d) − 1 = d/(1 − d) 28. a. 277.41, b. 247.69 29. PV = $61,796.71 30. Monthly mortgage payment = $1,119.71. Balance remaining after 5 years is $157,215. 35. Real rate of interest is zero. Annual consumption = $15,000. 41. n = 44.74 months 42. The present value of your payments is $671. The present value of your receipts is $579. This is a bad deal. 55. a. The present value of the payoff is $1,228. This is a good deal. b. PV is $771. This is a bad deal. 58. $4,126.57 59. a. $408,334.38, b. $3,457.40 63. a. 4%, b. 8.16%, c. 10.24% 65. a. $79.38, b. $91.51, c. 4.854%, d. $91.51/(1.04854)3 = $79.38 69. a. $261,139. b. $16,172 70. 18 years 71. Inflation = 1,099% per year 77. $1.188, $0.8418 Chapter 8 1. Both projects are worth pursuing. 3. NPVA = $11.93 and NPVB = $12.29. Choose B. 5. No. 7. Project A has a payback period of 2.5 years. Project B has a payback period of 2 years. 12. 2378 15. IRRA = 25.7% IRRB = 20.7% Project B is best. 16. NPV = −$197.7. Reject. 17. a. r = 0 implies NPV = $15,750. r = 50% implies NPV = $4,250. r = 100% implies NPV = 0. b. IRR = 100% 19. NPV9% = $2,139.28 and NPV14% = −$1,444.54. The IRR is 11.81%. 22. NPV must be negative. 24. a. Project Payback Discounted Payback A 3 4+ B 2 2.12 C 3 3.30 b. Project B c. Project B NPV A −1,011 B 3,378 C 2,405 e. False 26. a. NPV = $1,978 30. a. If r = 2%, choose A. b. If r = 12%, choose B. c. Larger cash flows for project A come later and are more sensitive to discount rate increases. 32. $22,774 34. b. At 5%, NPV = −$.443 c. At 20%, NPV = $.840 At 40%, NPV = −$.634 38. a. The equivalent cost of owning and operating the new machine is $4,590. The old machine costs $5,000 a year to operate. You should replace. b. If r = 10%, do not replace. Chapter 9 3. $2.2 million 5. Increase in net cash flow = $106 million 8. Cash flow = $3,300 10. Total operating cash flow in thousands (Years 1–6) = $339.45 Net cash flow at time 0 = −$1,000 11. a. Year CCA Rate 1 6,000 2 10,200 3 7,140 b. and c. See Solutions Manual. 16. After-tax cash flow = $17.3 million 18. a. Total incremental operating CF = $6,772 in years 1 − 6 Net cash flow at time 0 = −$4,000 b. NPV = $403.14 d. Project c. NPV = −4,000 + 1,300 × annuity factor (15%, 6 years) = $919.83; IRR = 23.21% 23. NPV = −10,894. Don’t buy. 25. Equivalent annual (net-of-tax) capital costs: Quick and Dirty: $2.12 million Do-It-Right: $1.84 million Choose Do-It-Right. 27. NPV = −$372,988 32. NPV = −$0.1817 million 34. c. NPV = $24.92 million IRR = 31.33% d. NPV (with CCA) = $9.21 million Chapter 10 1. Variable costs = $.50 per burger Fixed costs = $1.25 million 5. a. NPV = $5.1 million b. NPV = $2.5 million c. NPV = $6.2 million d. Price = $1.61 per jar 9. a. 4,286 diamonds annually b. 5,978 diamonds per year 11. Accounting break-even is unaffected. NPV break-even increases. 12. CF break-even is less than zero-profit break-even sales level. 15. a. Accounting break-even increases. b. NPV break-even falls. c. The switch to MACRS makes the project more attractive. 17. NPV will be negative. 20. DOL = 1 23. a. Average CF = 0 b. Average CF = $15,000 28. a. Expected NPV = −$681,728. The firm will reject the project. b. Expected NPV = $69,855. The project is now worth pursuing. Chapter 14 1. a. 80,000 shares b. Common shares $110,000 Retained earnings $ 30,000 Common equity $140,000 Note: Authorized shares $100,000 Issued shares $ 30,000 3. a. funded b. eurobond d. sinking fund f. prime rate g. floating rate h. private placement, public issue 6. a. 90 votes b. 900 votes 7. a. 200,001 shares b. 80,000 shares 11. Similarity: The firm promises to make specified payments. Advantage of income bonds: interest payments are taxdeductible expenses. Chapter 15 1. a. Subsequent issue b. Bond issue c. Bond issue 3. a. A large issue b. A bond issue After new issue: c. Private placements 4. Less underwriter risk; less signalling effect from debt; easier to value 7. a. 10% b. Average return = 3.94% c. I have suffered the winner’s curse. 10. No. 12. 12% of the value of funds raised. 15. a. $12.5 million b. $5.80 per share 17. a. $10 b. $18.333 c. $8.333 d. 200 rights Chapter 16 4. $296 million 11. P/E = 10/1.25 = 8 (no leverage) P/E = 10/1.33 = 7.5 (leveraged) 13. b. False c. True 17. requity = 18% 23. a. 11.27% b. Without the tax shield, the value of equity would fall by $296 million. Market-value balance sheet: Assets Liabilities and Equity 2,404 Debt 0 Equity 2,404 24. a. PV tax shield = $14 b. Assuming cost of debt = 8% and cost of equity = 15%, WACC = 12.55%. c. $150.47 25. Distorted investment decisions, impeded relations with other firms and creditors 36. a. Stockholders gain; bondholders lose. Saved cost 75,000 CCA tax −3,281 −5,742 −4,307 −3,230 −2,422 −7,267 shield Lease pay- −15,000 −15,000 −15,000 −15,000 −15,000 −15,000 ment Lease tax 5,250 5,250 5,250 5,250 5,250 0 shield Lease cash 61,969 −15,492 −14,057 −12,980 −12,172 −7,267 flows b. Equivalent loan = $53,292; NPV = $8,677 c. NPV = $1,017 13. a. $188 b. $1,471 c. $60 14. a. Minimum lease payment acceptable to lessor = $18,441.5 b. Maximum lease payment acceptable to lessee = $18,673 17. a. NPV to Nodhead = −$24,599 b. NPV to Compulease = $17,143 c. Overall gain = −$24,599 + $17,143 = −$7,456 Chapter 18 1. b. The ex-dividend date, June 7 c. Dividend yield = 1.1% d. Payout ratio = 15.8% e. New stock price = $24.55 3. a. Price = $32 b. Price = $32 c. Price = $40, unchanged 9. a. No effect on total wealth b. Identical to position after the stock repurchase 11. No impact on wealth 12. a. The after-tax dividend, $1.44 b. No 13. a. 1,250 shares. Value of equity remains at $50,000. b. Same effect as the stock dividend 14. a. $50 b. 26.16% c. $48.692 16. a. Price = $19.49 b. Before-tax return = 12.9% c. Price = $20.15 d. Before-tax return = 14.1% 26. a. $20 per share b. If the firm pays a dividend, EPS = $2. If the firm does the repurchase, EPS = $2.105. c. If the dividend is paid, the P/E ratio = 9.5. If the stock is repurchased, the P/E ratio = 9.5. Chapter 24 1. a. 65.13 euros; $153.55 b. 106.80 Swiss francs; $93.63 c. Direct exchange rate will decrease and indirect exchange rate will increase. d. U.S. dollar is worth more. 3. 1 + rx E(1 + ix) d. = 1 + r$ E(1 + i$) 6. a 8. Borrow the present value of 1 million Australian dollars, sell them for US dollars in the spot market, and invest the proceeds in an 8-year US dollar loan. In 8 years, it can repay the Australian loan with the anticipated Australian dollar payment. 10. a. 4.0% b. 10.5% c. −2.5% 11. 7.44% 14. Canadian dollar should be depreciating relative to the US dollar. 16. Net present value = $.72 million Chapter 26 6. Advantages: liquidity, no storage costs, no spoilage. Disadvantages: no income or benefits that could accrue from holding asset in portfolio. The benefit of futures is the ability to lock in a riskless position without paying any money. The benefit of the option hedge is that you benefit if prices move in one direction without losing if they move in the other direction. However, this asymmetry comes at a price: the cost of the option. Gold price 7. $1,330 $1,450 $1,650 a. Revenues $1,330,000 $1,450,000 $1,650,000 Futures contract 130,000 10,000 -190,000 b. Total $1,460,000 $1,460,000 $1,460,000 c. Revenues 1,330,000 $1,450,000 $1,650,000 +Put option payoff 120,000 0 0 −Put option cost 5,000 5,000 5,000 Net revenue $1,445,000 $1,445,000 $1,645,000 Petrochemical will take a long po its revenue from selling oil. sition to hedge its cost of buying oil. Onnex will take a short position to hedge Oil Price ($ per barrel) $88 $90 $92 Cost for Petrochemical: Cash flow on purchase of oil −88,000 −90,000 −92,000 + Cash flow on long futures position −2,000 0 +2,000 Total cash flow −90,000 −90,000 −90,000 Revenue for Onnex: Revenue from 1,000 barrels $88,000 $90,000 $92,000 + Payoff on short futures position 2,000 0 (2,000) Net revenue $90,000 $90,000 $90,000 The benefit of futures is the ability to lock in a riskless position without paying any money. The benefit of the option hedge is that you benefit if prices move in one direction without losing if they move in the other direction. However, this asymmetry comes at a price: the cost of the option. Solution Manual for Fundamentals of Corporate Finance Richard A. Brealey, Stewart C. Myers, Alan J. Marcus, Elizabeth Maynes, Devashis Mitra 9780071320573, 9781259272011
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