CHAPTER NINE Qualitative Questions 1. What is the difference between a call option and a put option? For an option buyer, a call option is the right to buy, while a put option is the right to sell. For an option writer, a call option is the obligation to deliver the asset to the holder at a specific price if the option is exercised while for a put the obligation is to buy the underlying asset if the option is exercised. A buyer buys a call in expectation that the market price will increase above the exercise price. A buyer buys a put in expectation that the market price will decrease below the exercise price. A writer sells an option hoping that it will expire worthless, leaving the writer with the option premium. A writer of a call option may already hold the underlying asset and want to sell at the agreed upon strike price. A writer of a put option may want to acquire the underlying assets. 2. What is the difference between a long position in a futures or forward contract and buying a call option? No cash is needed to buy futures/forwards, while buying an option requires payment of a premium. Each position earns a profit if the price of the underlying asset increases, though not in the same magnitude. A long futures/forward position produces profits as soon as the price of the underlying asset increases. The losses from buying a call option are limited to the call premium, while losses from a futures or forward position can be substantial. 3. What is the intrinsic value of an option? For call options, the intrinsic value is equal to the greater of zero or the market price of the asset minus the exercise price. For put options, the intrinsic value is equal to the greater of zero or the exercise price minus the market price of the asset. It is equal to the option value at the expiration of the option. 4. What determines the time value of an option? The market premium minus the intrinsic value The time to expiration The volatility of the return on the underlying asset The level of the risk-free rate The return on the underlying asset (e.g., dividend rate for shares, foreign interest rate for currency) 5. Why are common shares of a levered firm regarded as call options? Shareholders are entitled to any profits above the cash needed to service the debt, but they are subject to limited liability. Bondholders effectively own the firm and write a call option on the firm s profits. The exercise price is the present value of the debt. The option allows shareholders to buy back the firm from bondholders if the firm value exceeds the bond payment. 6. What is a standby agreement? The underwriter agrees to buy unsold shares in a rights offering at the subscription price or an agreed price. The underwriter agrees to buy unsold shares when the share issue is sold on a best efforts basis. 86 Solutions to Self-Test Questions
The standby agreement is a put option bought by the issuing firm. The issuing firm pays a premium to obtain the option. 7. Which corporate securities have call option qualities? convertibles extendibles warrants 8. Why do firms issue securities with options? The firm s securities are high risk. General market conditions are not good for new share issues. The market for the firm s securities is depressed. Options promise participation in the future profitability of the firm. 9. What are the advantages of using an option to hedge? Buying options limits potential losses if rates move adversely and allows the hedger to gain if rates move favourably. Buying options is similar to buying insurance because the option buyer must pay a premium for this protection. 10. What is a swap? It is an agreement between two parties to exchange future cash flows. In a currency swap, there is an exchange of two or more liabilities denominated in different currencies. In an interest-rate swap, a liability offering a floating interest rate is exchanged for a liability offering a fixed rate. 11. How can a currency swap hedge foreign-exchange treasury risk? When a company C X has income denominated in currency X and debt obligations denominated in currency Y, a currency swap can hedge the risk. The risk is that currency Y may appreciate over currency X, requiring more income in currency X to service the debt. The swap would involve another company C Y that has income denominated in currency Y and debt obligations denominated in currency X. With the swap, C Y would service the debt for C X, and C X would service the debt for C Y. 12. What are the benefits and costs of hedging? Benefits include securing a selling or buying price, guaranteeing the cost of capital, and delaying the purchase of a commodity until it is physically needed. Costs include transaction costs, costs of hedging, the need for credit facilities for overthe-counter hedging, costs of margin requirements for exchange-traded hedging, and administrative costs. The administrative costs vary depending on the hedging approach taken by the hedger. Qualitative Multiple Choice Questions Question 1 iv) Similarly to trading in futures, trading in exchange-traded options is managed by a clearing house to reduce traders risk. Question 2 ii) They are equal to the time value of the option if the market value of the underlying asset is greater than the exercise price. Question 3 iii) They can be conversion privileges on bonds that are issued when the stock market is considered temporarily depressed. Solutions to Self-Test Questions 87
Question 4 i) The market price of the common share Question 5 i) Buy a call option Question 6 i) Buy call options Question 7 i) The market price of the underlying asset is above the exercise price Question 8 ii) Buy a call Question 9 iv) Buy call options Quantitative Multiple Choice Questions Question 1 iii) Put value = $4.55 + $40 (e -0.05 0.5 ) - $36 = $4.55 + $39.01 - $36 = $7.56 Question 2 iii) Question 3 iv) Question 4 iii) 1 Principal = $10 million * $1.569 = 6,373,486 6,373,486 12% interest = 764,818 Euro Time value = option premium - (share price - exercise price) = $4.26 - ($40 - $38) = $2.26 Value of put at expiration = exercise price - share price = $60 - $46 = $14 Profit per share = value of put - premium paid = $14 - $4 = $10 per share Question 5 ii) The potential saving in interest costs is 1% (1/2% for each party). MNO Corp. PQR Corp. Quality spread Fixed-rate market 12% 10% 2% Floating-rate market Prime + 1% Prime 1% Net quality differential 1% 88 Solutions to Self-Test Questions
PQR borrows in the floating-rate market so its net payment should be prime minus 1/2% after the swap; that is, PQR will pay the following rate: In fixed-rate market, pay 10% In the swap deal, pay Prime In the swap deal, receive X Net payment Prime - 1/2% Prime - 1/2% = 10% + Prime - X X = 10.5%, which is the fixed rate MNO should pay in the swap. To check, MNO: In floating-rate market, pay Prime + 1% In the swap deal, pay 10.5% In the swap deal, receive Prime Net payment = Prime + 1% + 10.5% - Prime = 11.5%, which is 1/2% less than 12% the fixed rate MNO has to pay if it borrows long-term fixed rate outright without the swap deal. Question 6 ii) Gain = 100(strike price - market price) - premium paid = 100($45 - $35) - $54 = $946 Question 7 iii) Value of put at expiration = (exercise price - share price)(number of shares) = ($13 - $6)(100) = $700 Profit from put investment = value of put - premium paid $ 600 = $700 - $100 Loss from stock investment = $6(100) - $10(100) (400) Total profit $ 200 Alternatively, her total profit = (selling price - buying price - option premium)(number of shares) = ($13 - $10 - $1)(100) = $200. Quantitative Problems Problem 1 In this question, you must demonstrate your knowledge of how interest rate swaps can be used to benefit both parties. The factors that affect the magnitude of the benefits must be explained in an internal management memo. You must structure your answers in a way that demonstrates your understanding of underlying concepts, your communication skills, and your ability to apply book materials to a situation where owners and managers have particular preferences. Internal Memo Date: June 16, 20X8 To: Samantha Black, Treasurer From: Arnold Schultz, Assistant Subject: Report on Interest Rate Swap Opportunity Interest Rates Because of the differential creditworthiness of our company and Company X, the loan rates the two companies can obtain in the fixed- and floating-rate markets differ. Both rates are more favourable for us, as the lower-risk firm. The fixed rate is relatively more favourable for us, as shown in the following table. Solutions to Self-Test Questions 89
Interest Rate Swap Opportunity Lower-risk firm Higher-risk firm (Anderson) (Company X) Quality spread Fixed-rate 7% 10.5% 3.5% market Floating-rate BA rate + 1.5% BA rate + 3% 1.5% market Net quality differential 2.0% Interest-Rate Swap Agreement A swap can be arranged on the underlying loan principal of $10 million. For the remaining $5 million, we can borrow fixed, borrow directly on the floating-rate market, or swap with another party. The swap described here relates only to the $10 million match for our company and Company X. A swap would save 2%($10 million) = $200,000 per year. Some of this amount would be used to compensate the investment banker, and then we would share the net savings with Company X. A swap would involve us, as the lower-risk firm, borrowing where we have a comparative advantage that is, in the fixed-rate market and then swapping interest payments with Company X. Each borrower would then obtain financing of the type they prefer (fixed or floating rate) and would effectively pay a lower rate than if they had borrowed on their own account. The source of the savings to swap participants is the net quality differential, caused by the difference in credit risk of the two borrowers. It is higher when the fixed-rate loan has longer maturity. In effect, we would get the benefit of our longstanding low-risk strategy, since this is the reason we can borrow on favourable terms, generating the net quality differential. We will receive the greater portion of the gain because we have a better (i.e., lower) credit risk, with the remainder of gain available to Company X to entice it to participate in the swap. This possible benefit must be weighed against management s possible preference to borrow on a fixed-rate basis for financing our long-term asset acquisition. The opportunity for a swap may not alter this decision, but you can inform management of the available benefits of this arrangement. Problem 2 MEMO Date: April 15, 20X8 To: Jean Sawyer, CEO From: Antonio Bing, CGA Subject: Alternatives for Exchange Risk The expected Canadian dollar cost of each hedging alternative is as follows: 1) Forward hedge: You would buy a forward contract. Though you don t need to pay for the contract right now, 6 months later you will have to pay the Canadian dollar cost of the forward hedge, which is: 200,000a C$2.47 b = C$494,000 1 90 Solutions to Self-Test Questions
2) Money market hedge: For this alternative, you would borrow Canadian dollars and convert these dollars into British pounds at today s spot rate. Then you would deposit these pounds in a British bank to earn interest. Because your deposit earns interest in British pounds, the amount of British pounds you would need today is: 200,000 1 + 4.5% * 6 = 195,599 12 To get this amount of British pounds at today s spot rate, you would need to borrow: 195,599a C$2.48 b = C$485,085.52 1 Since you borrowed this amount of Canadian dollars, you would have to pay interest. The Canadian dollar cost of this alternative is: C$485,085.52a1 + 5% * 6 12 b = C$497,212.66 3) Option hedge: You would purchase a call on pounds. The maximum Canadian dollar cost of this alternative is: 200,000(C$2.45 + C$0.05) = C$500,000 One benefit unique to a purchase call option hedge is you retain the potential to reduce your cost if the exchange rate moves in your favour, that is, if the Canadian dollar appreciates against British pounds in this case. You may pay fewer Canadian dollars to get the same amount of pounds. $500,000 is the maximum cost to you. 4) Remaining unhedged: The Canadian dollar cost of this alternative cannot be determined now, as it depends on the spot exchange rate 6 months later, which is unpredictable. In general, the Canadian dollar cost of remaining unhedged is: 200,000 (spot rate 6 months later) Recommendation I would recommend the option hedge alternative. Although remaining unhedged allows you to benefit from a lower value of pounds, it is more risky because it is equally likely that the British pound will strengthen against the Canadian dollar. Both forward and money market hedges eliminate the risk of a strong pound but they also eliminate your opportunity to benefit from a weak pound. An option hedge provides you with both protection against a weaker Canadian dollar and potential to benefit from a stronger Canadian dollar. Problem 3 In this question, an investor has a portfolio containing shares and put options on the same shares. He is contemplating participating in a forward agreement to purchase more shares in the company. a) i) A put on shares is an option that gives that put holder a right to sell or an obligation to the put seller to buy at a specific price, up to a specific date. ii) The value of a put option is affected: positively by the exercise price positively by the time to expiry negatively by the current interest rate positively by the variability or beta in the underlying share price negatively by the price of the underlying shares Solutions to Self-Test Questions 91
b) A forward agreement is a contract (i.e., an obligation) for the seller to purchase and for the holder to sell the share on a set future date at a set price. c) The overall profit/loss on Jimmy s portfolio under each scenario for future share price is as follows: i) If the share price is $40 Gain on cash position [250 ($40 - $35)] $ 1,250 Put option premium paid [500($1)] (500) Profit, without the forward 750 Gain on forward contract [100 ($40 - $33)] 700 Overall profit $ 1,450 ii) If the share price is $25 Gain on option [500($30 - $25)] $ 2,500 Loss on cash position [250 ($25 - $35)] (2,500) Put option premium paid [500($1)] (500) Loss, without the forward (500) Loss on forward contract [100 ($25 - $33)] (800) Overall loss $(1,300) If Jimmy does not have the forward contract when the share price rises to $40, the profit would be only $750, and when the share price falls to $25 the loss would be only $500. There would be less variability in the returns on his portfolio if he did not participate in the forward agreement. Problem 4 a) Using the Black-Scholes option-pricing model, the first step is to calculate d 1 using Equation 9-5: d 1 = [ln(s/e) + rt], (st 1/2 ) + (st 1/2, 2) = [ln($12/$10) + (0.05 0.326)], (0.12 0.326 1/2 ) + [(0.12 0.326 1/2 ), 2] = 2.9332 Next, calculate d 2 = d 1 - st 1/2 = 2.9332 - (0.12 0.326 1/2 ) = 2.9332-0.0685 = 2.8647 Then using Equation 9-6 to calculate the values of N(d 1 ) and N(d 2 ), you will get: N(d 1 ) = 0.9983 + [(2.9332-2.93), (2.94-2.93)] [0.9984-0.9983] = 0.9983 + [(0.0032, 0.01) 0.0001] = 0.99833 N(d 2 ) = 0.9979 + [(2.8638-2.86), (2.87-2.86)] [0.9979-0.9979] = 0.9979 Finally, you calculate the call premium using Equation 9-4, as follows: C = SN(d 1 ) - EN(d 2 )e -rt = ($12)(0.99833) - ($10)(0.9979)(2.718282) (-0.05)(0.326) = $2.16 b) The value of the option on the expiry date = maximum (0, market price of share - exercise price). In this case, the option would be worth $12 - $10 = $2.00. 92 Solutions to Self-Test Questions
c) As in part b), the option value on expiry date is maximum (0, market price - exercise price). In the given case, the value of the option would be zero because the exercise price is greater than the market price of the shares. Problem 5 a) CTR is buying yen in the market, while JTD is selling yen in the market. A foreign exchange dealer will be willing to buy at the bid price and sell at the ask price. At a given point in time, the ask price is normally greater than the bid price to allow a profit margin for the dealer. The current ask-bid spread is 2 per Canadian dollar. b) Both companies should use a currency swap agreement to avoid transaction costs caused by their need for either Canadian dollars or Japanese yen. Each company must exchange currencies for transactions such as initial investment outlay, reversal of this investment outlay when the project is terminated, and any debt service requirements during the project. For example, JTD has debt denominated in yen, which will be used in Canada. The company has to exchange the yen for Canadian dollars to pay for its acquisition plans. Once JTD acquires some Canadian businesses, the revenues generated from the investment will be in Canadian dollars, out of which the debt obligations must be serviced. Therefore, at every coupon payment date, the company needs to exchange Canadian dollars for yen at the spot price to service the debt. CTR is in the opposite situation. It has debt denominated in Canadian dollars, which will be used in Japan. The company has to exchange the dollars for yen to pay for its expansion plans. The revenues from the Japanese operations will be in yen. Some of these revenues will be used to service the debt obligations. Therefore, at every coupon payment date, the company needs to exchange Japanese yen for Canadian dollars. c) Both JTD and CTR will incur savings from transaction costs caused by the exchange of currencies. Savings to JTD Without the swap agreement, JTD will have to exchange the initial investment outlays of 2.52 billion for Canadian dollars. This transaction will have to be reversed after five years. Moreover, for five years, JTD will have to exchange dollars for yen to service the debt requirements. Debt service requirements: Per year: 2.52 billion 0.03 = 75.6 million Five years: 75.6 million 5 = 378 million Total exchange over five years: ( 2.52 billion 2) + 0.378 billion = 5.418 billion Savings to JTD: a 5.418 billion b * 2 84 dollar = 129 million (or $1.536 million) Savings to CTR Similarly, without the swap agreement, CTR will have to exchange the initial investment outlays of $30 million for yen. This transaction will have to be reversed after five years. Moreover, for five years, CTR will have to exchange yen for Canadian dollars to service the debt requirements. Solutions to Self-Test Questions 93
Debt service requirements: Per year: $30 million 0.05 = $1.5 million Five years: $1.5 million 5 = $7.5 million Total exchange over five years: ($30 million 2) + $7.5 million = $67.5 million Savings to CTR: $67.5 million 2/dollar = 135 million (or $1.607 million) d) The exchange of the initial investment outlays will not lead to losses or profits. Assuming an exchange rate of 84 per dollar makes 2,520 million equal to $30 million. The first table shows that, for year 1, JTD will incur losses of $0.555 million ($0.945 - $1.5) because of the use of the swap. This table also shows the losses incurred from years 2 to 5. In contrast, the second table shows that, for year 1, CTR will incur profits of $0.555 million ($1.5 - $0.945) because of the use of the swap. The table also shows the profits incurred from years 2 to 5. Swap Net Cash Flows to JTD Cash outflows Outflows from use of swap without Swap cash End of Exchange swap flows year ( millions) rate ($ millions) ($ millions) ($ millions) 1 75.6 80 $ 0.945 $ 1.5 $(0.555) 2 75.6 84 0.900 1.5 (0.600) 3 75.6 86 0.879 1.5 (0.621) 4 75.6 82 0.922 1.5 (0.578) 5 75.6 77 0.982 1.5 (0.518) Coupon 5 2,520.0 77 32.727 30.0 2.727 Principal Swap Net Cash Flows to CTR Outflows Cash outflows from use of swap without Swap cash End of Exchange swap flows year ( millions) rate ($ millions) ($ millions) ($ millions) 1 75.6 80 $ 0.945 $ 1.5 $ 0.555 2 75.6 84 0.900 1.5 0.600 3 75.6 86 0.879 1.5 0.621 4 75.6 82 0.922 1.5 0.578 5 75.6 77 0.982 1.5 0.518 Coupon 5 2,520.0 77 32.727 30.0 (2.727) Principal 94 Solutions to Self-Test Questions
Problem 6 a) With options, only the writer is required to post collateral in the form of margin requirements. Since the option holder has no liability, there is no need for margin requirements. The writer s margin account is marked to market on a daily basis. At expiration, the writer s margin account reflects the updated losses or profits. For example, suppose the option is a put on a stock with an exercise price of $15. On the expiry date, the share price is $10. The writer is obligated to buy the asset at the exercise price. This means a loss of $5 per share. The writer s margin account is already debited $5 per share, which is now held in trust for the put holder. The put holder can accept the $5 per share in cash settlement and sell the amount of shares covered by the put in the cash market for $10 per share. The total proceeds from this alternative are $15 per share. Alternatively, the holder can demand delivery. The shares are delivered to the clearing house, which sells them for $10 per share. The $5 per share held in trust for the holder is added to the $10 per share from the sale to make up the $15 promised to the put holder. b) Differences between forward and options contracts include: Options have up-front premiums and forwards do not. Options allow for upside potential while hedging downside risk, whereas forward contracts provide a certain outcome regardless of changes in market rates and prices. Problem 7 a) i) GA is already paying T-bill rate + 1.5% in the floating-rate market. MM should borrow $10 million in the fixed-rate market. ii) They should enter into an interest rate swap, such that the principal is $10 million, the term of the swap is 5 years, and the floating rate under the swap, is the T-bill rate, which MM pays to GA. iii) Through the swap, GA should get fixed-rate financing at 0.5% less than it could directly, that is, at the fixed rate of 8% - 0.5% = 7.5%. iv) The fixed rate under the swap can then be determined. Let x be the fixed rate under the swap, which GA pays to MM. The effective rate paid by GA is T-bill rate + 1.5% + x - T-bill rate = 7.5%. Thus, x = 6%. v) The effective rate paid by MM is 6% + T-bill rate - 6% = T-bill rate, which is 1% less than T-bill rate + 1%, which MM could get directly in the floating-rate market. vi) MM might believe that interest rates have already peaked and are going to decline soon. Using floating-rate financing enables MM to benefit from decreasing interest rates. b) i) If the commercial paper were issued today, the amount of capital that GA would have raised would be: $10,000,000 1 + 6% * 9 12 ii) The number of BA futures contracts GA should sell would be: 0.9 * $9,569,378 * 9 12 $1,000,000 * 3 12 = $9,569,378.00 = 25.8, rounded to 26. iii) On the refinancing day 6 months later, the amount of capital that GA could raise from issuing new commercial paper would be: $10,000,000 1 + 8.5% * 9 = $9,400,705.00 12 Solutions to Self-Test Questions 95
If the futures price for BA falls from 98.5 to 96.2, a fall of 2.3% or 230 basis points, GA s profit on the futures position would be: 26 230 $25 = $149,500 The effective price GA would receive for the new commercial paper is: $9,400,705 + $149,500 = $9,550,205 The interest expense paid by GA over 9 months would be: The effective interest rate paid by GA is: The effective annual rate is: $10,000,000 - $9,550,205 = $449,795 $449,795 $9,550,205 = 4.71% (1 + 4.7%) 12/9-1 = 6.33% Cases Case 1: Allied Business Corporation (ABC) This case involves application of the Black-Scholes option-pricing model to value a call and using put-call parity to value the corresponding put. The outcome for an investor with a combination of holdings must also be calculated. a) Current share price = $33 Exercise price = $30 Risk-free rate = 5% per annum Time to maturity = 6 months = 0.5 year Standard deviation of share returns = 12% cina 33 30 b + 0.05(0.5) d d 1 = + 0.12(0.5)0.5 0.12(0.5) 0.5 2 [0.0953 + 0.025] = + 0.08485 = 1.418 + 0.042425 = 1.46 0.08485 2 d 2 = 1.46-0.12(0.5) 0.5 = 1.46-0.08485 = 1.37515 Call value = $33 N(1.46) - $30 N(1.38)e-0.05 (0.5) = $33(0.9279) - $30(0.9162)0.9753 = $30.62 - $26.81 = $3.81 b) Using put-call parity to value the corresponding put option: Put value = Call value + $30 e -0.05(0.5) - $33 = $3.81 + $30(0.9753) - $33 = $3.81 + $29.26 - S33 = $0.07 The put is worth $0.07. c) The values of put options are: negatively related to share price positively related to exercise price 96 Solutions to Self-Test Questions
positively related to the time remaining to expiry negatively related to the current interest rate positively related to the variability in the share price d) There is a 25% chance that the share price will be $30, according to Jack. i) There is a 25% chance the share price will be $30, in which case: The 200 shares are worth 200(30) = $6,000. The puts have a strike price of $30 so they expire worthless. A forward agreement allowing the sale of 100 shares at $33 will provide for $3 over the market value on 100 shares for a gain of $300. Total position value = $6,300 at end of 6-month period ii) There is a 40% chance the share price will be $33, in which case: The 200 shares are worth 200(33) = $6,600. The puts have a strike price of $30 so they expire worthless. A forward agreement allowing the sale of 100 shares at $33 will provide no difference compared with the market value. Total position value = $6,600 at end of 6-month period iii) There is a 35% chance the share price will be $35, in which case: The 200 shares are worth 200(35) = $7,000. The puts have a strike price of $30 so they expire worthless. A forward agreement allowing the sale of 100 shares at $33 will result in a loss of $2 compared with the market value on 100 shares for a loss of $200. Total position value = $6,800 at end of 6-month period Expected position = 0.25(6,300) + 0.4(6,600) + 0.35(6,800) = 1,575 + 2,640 + 2,380 = $6,595 Case 2: YYY Corporation In this case, use and interpret the Black-Scholes option-pricing model and also the put-call parity relationship. A call and a corresponding put are valued. The time value of the option is also analyzed in some detail. a) The Black-Scholes option-pricing model generates the following value for this call option: Share price = S = $40 Strike price = E = $38 Risk-free rate = r = 3.5% Time to expiry = T = 75 365 = 0.2055 years Variance of returns = 0.14 Standard deviation of returns = s = 0.3742 d 1 = = Ina S E b + rt + st1/2 2 st 1/2 cina 40 38 b + (0.035)(0.2055) d + (0.3742)(20.2055) (0.3742)( 20.2055) 2 Solutions to Self-Test Questions 97
d 2 = d 1 - st 1/2 = 0.42958 - (0.3742)(0.45332) = 0.42958-0.16963 = 0.25995 or 0.26 Probability density tables for the normal distribution must be used to calculate N(d 1 ) and N(d 2 ) as follows: d 1 is 0.43 for a table value for N(d 1 ) = 0.6664 d 2 is 0.26 for a table value for N(d 2 ) = 0.6026 The call value can now be calculated as follows: = 26.656-22.899e -0.035(0.2055) = 26.656-22.899(0.99283) = 26.656-22.735 = $3.92 b) The time value of a call option is the value of having time before the option expires for the share price to change. The share price could increase further, making the payout at option exercise more valuable for the call owner. For this call option, the value of exercising it immediately is the current difference between the market price and the strike price, or $40 - $38 = $2. The value of the option is higher than this. It is worth $3.92. Time value = option value - value of immediate exercise = $3.92 - $2.00 = $1.92 The greater the time to maturity, the higher the time value. If this option had 150 days to expire rather than 75, the value of the option using the Black-Scholes model could be recalculated as follows: 150 Now T = = 0.41096 365 Ina S E b + rt d 1 = [0.05129 + 0.0071925] = + (0.3742)(0.45332) (0.3742)(0.45332) 2 = 0.34476 + 0.08482 = 0.42958 st 1/2 Call value = $40(0.6664) - $38(0.6026)e - rt + st1/2 2 cina 40 38 b + (0.035)(0.41096) d = + (0.3742)(20.41096) (0.3742)( 20.41096) 2 [0.05129 + 0.01438] = + (0.3742)(0.64106) (0.3742)(0.64106) 2 = 0.27376 + 0.1199 = 0.3937 or approximately 0.39 d 2 = d 1 - st 1/2 = 0.3937 - (0.3742)(0.64106) = 0.3937-0.2399 = 0.1538 or 0.15 Probability density tables for the normal distribution must be used to calculate N(d 1 ) and N(d 2 ) as follows: d 1 is 0.39 for a table value for N(d 1 ) of 0.1517; add 0.5 to get 0.6517 d 2 is 0.15 for a table value for N(d 2 ) of 0.0596; add 0.5 to get 0.5596 98 Solutions to Self-Test Questions
Call value can now be calculated as follows: = $40(0.6517) - $38(0.5596)e -rt = 26.068-21.265e -0.035(0.41096) = 26.068-21.265 (0.98572) = 26.068-20.961 = $5.11 Time value = option value - value of immediate exercise = $5.11 - $2.00 = $3.11 The time value of the option increases significantly, to $3.11. c) Using put-call parity, we can determine the value of the corresponding put as follows: P = C + E(e -rt ) P = $3.92 + $38(e -0.035(0.2055) ) - $40 = $3.92 + $38 (0.99283) - $40 = $1.65 Lucille will purchase a put if she thinks the price of the shares will go down below the strike price. She will purchase a call if she thinks the share price will increase to a level above the strike price. Case 3: JKL Corporation Risks JKL is exposed to commodity (gold) price risk, foreign-exchange risk, and interest-rate risk. Gold is JKL s production input. The increasing gold price reduces its profit because JKL needs to pay U.S. dollars to purchase gold while selling its products for Canadian dollars. By using short-term funds, JKL has to continually renew its financing and pay floating interest rates, and short-term interest rates are volatile. Hedging Vehicles Forward/futures contracts, options, and currency and interest-rate swaps are available to JKL in hedging its risks. For the gold-price risk, JKL may buy futures contracts to hedge at a specified strike price. As each contract entitles the buyer to buy 100 ounces of gold and JKL plans to buy 5,000 5,000 ounces, it must purchase 100 = 50 contracts. One month later, if the gold price increases as expected, JKL has to pay a higher price for gold on the spot market, incurring a loss; however, it will make a profit in its futures position. With a loss from the cash market and a profit from the futures market cancelling each other, JKL essentially locks in the fixed strike gold price. If the gold price decreases, JKL will benefit from the low spot price, which will be offset by the loss in futures position. Regardless of which direction gold price moves, JKL is protected from a volatile gold price. For foreign-exchange risk, JKL should sell currency futures contracts on Canadian dollars. Assuming a gold price of US$450 per ounce and an exchange rate of C$1.20 = US$1.00, JKL C$1.20 C$100,000 should sell 5,000 ounces US$450 = 27 contracts. JKL could also buy currency futures contracts on U.S. dollars. Futures contracts on currencies work in the same way as gold futures contracts. If JKL wants to use currency options, it should buy put options on Canadian dollars or buy call options on U.S. dollars. If the Canadian dollar appreciates against the U.S. dollar, JKL will let the options expire and buy U.S. dollars on the spot market using Canadian dollars. On the other hand, if the Canadian dollar depreciates against the U.S. dollar, JKL will exercise its options and receive a higher price for its Canadian dollars or pay a lower price for U.S. dollars. Solutions to Self-Test Questions 99
With interest-rate risk, JKL may sell BA futures contracts. This type of contract works as follows: If the commercial paper were issued today, the amount that JKL could raise by the issue of 9-month commercial paper with a face value of $4 million and a yield of 4% would be: $4 million = $3,883,495.15 1 + 0.04 * 9 12 The number of futures contracts JKL should sell would be: 0.9 * $3,883,495.15 * 9 12 $1,000,000 * 91 365 = 10.51 rounded to 11 Three months later, if the 9-month commercial paper trades to yield 6%, JKL can issue the commercial paper with a face value of $4 million, raising: $4,000,000 = $3,827,751.20 If the futures price for BA falls from 97 to 94, a fall of 3% or 300 basis points, JKL s profit on the futures position will be: 11 300 $25 = $82,500 The effective price JKL would receive for the commercial paper is: $3,827,751.20 + $82,500 = $3,910,251.20 The interest expense paid by JKL over 9 months is: $4,000,000 - $3,910,251.20 = $89,748.80 The effective interest rate paid by JKL is: Effective annual rate: 1 + 0.06 * 9 12 $89,748.80 $3,910,251.20 = 2.2952% (1 + 2.2952%) 12/9-1 = 3.01% Comparison of Hedging Vehicles Forward and futures contracts are essentially the same, both fixing the price of an asset to be delivered on a future date. But forward contracts are customized, while futures contracts are standardized. In this case, JKL is able to purchase a forward contract for 2.7 million Canadian dollars (= 5,000 ounces US$450 C$1.20/$100,000). If it wants to use futures, it has to purchase 27 contracts. Options are more flexible than forward/futures contracts. They provide their users with downside protection and at the same time keep upward profitable potential for their uses. But options are not free. Alternative Strategies It is possible for JKL to hedge its risks without using derivatives but through its operations. For example, JKL may invoice its customers in U.S. dollars to transfer the foreign exchange risk to its customers. JKL may also use long-term funds to reduce the interest rate risk. Or JKL may expand its business to the United States. 100 Solutions to Self-Test Questions