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Derivatives Markets 4th edition Rober L. Macdonald - Solutions
If XYZ does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copper production, what is its estimated profit 1 year from now? Construct graphs illustrating both unhedged and hedged profit.
Compute estimated profit in 1 year if Telco sells collars with the following strikes:a. $0.95 for the put and $1.00 for the call.b. $0.975 for the put and $1.025 for the call.c. $0.95 for the put and $0.95 for the call.Draw a graph of profit in each case.
Compute estimated profit in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero): a. Sell one 0.975-strike call and buy two 1.034-strike calls. b. Sell two 1.00-strike calls and buy three 1.034-strike calls.
Suppose that Wirco does nothing to manage the risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose that Wirco buys copper forward at $1. What is its profit 1 year from now?
What happens to the variability of Wirco's profit if Wirco undertakes any strategy (buying calls, selling puts, collars, etc.) to lock in the price of copper next year? You can use your answer to the previous question to illustrate your response.
Golddiggers has zero net income if it sells gold for a price of $380. However, by shorting a forward contract it is possible to guarantee a profit of $40/oz. Suppose a manager decides not to hedge and the gold price in 1 year is $390/oz. Did the firm earn $10 in profit (relative to accounting
Consider the example in Table 4.6. Suppose that losses are fully tax-deductible. What is the expected after-tax profit in this case?
Suppose that firms face a 40% income tax rate on all profits. In particular, losses receive full credit. Firm A has a 50% probability of a $1000 profit and a 50% probability of a $600 loss each year. Firm B has a 50% probability of a $300 profit and a 50% probability of a $100 profit each year.a.
Suppose that firms face a 40% income tax rate on positive profits and that net losses receive no credit. (Thus, if profits are positive, after-tax income is (1− 0.4) × profit, while if there is a loss, after-tax income is the amount lost.) Firms A and B have the same cash flow distribution as in
a. Suppose that Auric insures against a price increase by purchasing a 440-strike call. Verify by drawing a profit diagram that simultaneously selling a 400 strike put will generate a collar. What is the cost of this collar to Auric?b. Find the strike prices for a zero-cost collar (buy high-strike
Suppose that LMN Investment Bank wishes to sell Auric a zero-cost collar of width 30 without explicit premium (i.e., there will be no cash payment from Auric to LMN). Also suppose that on every option the bid price is $0.25 below the Black Scholes price and the offer price is $0.25 above the
Suppose the 1-year copper forward price were $0.80 instead of $1. If XYZ were to sell forward its expected copper production, what is its estimated profit 1 year from now? Should XYZ produce copper? What if the forward copper price is $0.45?
Use the same assumptions as in the preceding problem, without the bid-ask spread. Suppose that we want to construct a paylater strategy using a ratio spread. Instead of buying a 440-strike call, Auric will sell one 440-strike call and use the premium to buy two higher-strike calls, such that the
Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coefficient of about 100,000.
Using the information in Table 4.9 about Scenario C:a. Compute σtotal revenue when correlation between price and quantity is positive.b. What is the correlation between price and revenue?
Using the information in Table 4.9 about Scenario C:a. Using your answer to the previous question, use equation (4.7) to compute the variance-minimizing hedge ratio.b. Run a regression of revenue on price to compute the variance-minimizing hedge ratio.c. What is the variability of optimally hedged
Using the information in Table 4.9 about Scenario C:a. What is the expected quantity of production?b. Suppose you short the expected quantity of corn. What is the standard deviation of hedged revenue?
Suppose that price and quantity are positively correlated as in this table:There is a 50% chance of either price. The futures price is $2.50. Demonstrate the effect of hedging if we do the following:a. Short the expected quantity.b. Short the minimum quantity.c. Short the maximum quantity.d. What
Compute estimated profit in 1 year if XYZ buys a put option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
Compute estimated profit in 1 year if XYZ sells a call option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
Compute estimated profit in 1 year if XYZ buys collars with the following strikes:a. $0.95 for the put and $1.00 for the call.b. $0.975 for the put and $1.025 for the call.c. $1.05 for the put and $1.05 for the call.Draw a graph of profit in each case.
Compute estimated profit in 1 year if XYZ buys paylater puts as follows (the net premium may not be exactly zero):a. Sell one 1.025-strike put and buy two 0.975-strike puts.b. Sell two 1.034-strike puts and buy three 1.00-strike puts.
If Telco does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper that it buys? If it hedges the price of wire by buying copper forward, what is its estimated profit 1 year from now? Construct graphs illustrating both unhedged and hedged profit.
Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
Compute estimated profit in 1 year if Telco sells a put option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
Construct Table 5.1 from the perspective of a seller, providing a descriptive name for each of the transactions.
The S&R index spot price is 1100 and the continuously compounded risk-free rate is 5%. You observe a 9-month forward price of 1129.257.a. What dividend yield is implied by this forward price?b. Suppose you believe the dividend yield over the next 9 months will be only 0.5%. What arbitrage would
Suppose the S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin.a. What is the notional value of your position?b. Assuming a 10% initial margin, what is the value of the initial margin?
Suppose the S&P 500 index is currently 950 and the initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts.a. What is the notional value of your position? What is the margin?b. Suppose you earn a continuously compounded rate of 6% on your margin balance, your position is
Verify that going long a forward contract and lending the present value of the forward price creates a payoff of one share of stock whena. The stock pays no dividends.b. The stock pays discrete dividends.c. The stock pays continuous dividends. Discuss.
Verify that when there are transaction costs, the lower no-arbitrage bound is given by equation (5.12).S0 × P5.5 + 0.7 × BSCall(S0, S0, σ , r, 5.5, δ)
Suppose the S&R index is 800, and that the dividend yield is 0. You are an arbitrageur with a continuously compounded borrowing rate of 5.5% and a continuously compounded lending rate of 5%. Assume that there is 1 year to maturity.a. Supposing that there are no transaction fees, show that a
Suppose the S&P 500 currently has a level of 875. The continuously compounded return on a 1-year T-bill is 4.75%. You wish to hedge an $800,000 portfolio that has a beta of 1.1 and a correlation of 1.0 with the S&P 500.a. What is the 1-year futures price for the S&P 500 assuming no
Suppose you are selecting a futures contract with which to hedge a portfolio. You have a choice of six contracts, each of which has the same variability, but with correlations of −0.95, −0.75, −0.50, 0, 0.25, and 0.85. Rank the futures contracts with respect to basis risk, from highest to
Suppose the current exchange rate between Germany and Japan is 0.02 = C/¥. The euro-denominated annual continuously compounded risk-free rate is 4% and the yen-denominated annual continuously compounded risk-free rate is 1%. What are the 6-month euro/yen and yen/euro forward prices?
Suppose the spot $/¥ exchange rate is 0.008, the 1-year continuously compounded dollar-denominated rate is 5% and the 1-year continuously compounded yen denominated rate is 1%. Suppose the 1-year forward exchange rate is 0.0084. Explain precisely the transactions you could use (being careful
A $50 stock pays a $1 dividend every 3 months, with the first dividend coming 3 months from today. The continuously compounded risk-free rate is 6%.a. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend?b. What is the price
Suppose we wish to borrow $10 million for 91 days beginning next June, and that the quoted Eurodollar futures price is 93.23.a. What 3-month LIBOR rate is implied by this price?b. How much will be needed to repay the loan?
A $50 stock pays an 8% continuous dividend. The continuously compounded riskfree rate is 6%.a. What is the price of a prepaid forward contract that expires 1 year from today?b. What is the price of a forward contract that expires at the same time?
Suppose the stock price is $35 and the continuously compounded interest rate is 5%.a. What is the 6-month forward price, assuming dividends are zero?b. If the 6-month forward price is $35.50, what is the annualized forward premium?c. If the forward price is $35.50, what is the annualized continuous
Suppose you are a market-maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.a. What is the no-arbitrage forward price for delivery in 9 months?b. Suppose a customer wishes to enter a short index futures
Repeat the previous problem, assuming that the dividend yield is 1.5%.a. What is the no-arbitrage forward price for delivery in 9 months?b. Suppose a customer wishes to enter a short index futures position. If you take the opposite position, demonstrate how you would hedge your resulting long
The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.a. Suppose you observe a 6-month forward price of 1135. What arbitrage would you undertake?b. Suppose you observe a 6-month forward price of 1115. What arbitrage would you undertake?
The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous dividend yield on the index is 2%.a. Suppose you observe a 6-month forward price of 1120. What arbitrage would you undertake?b. Suppose you observe a 6-month forward price of 1110. What arbitrage would you undertake?
Suppose that 10 years from now it becomes possible for money managers to engage in time travel. In particular, suppose that a money manager could travel to January 1981, when the 1-year Treasury bill rate was 12.5%.a. If time travel were costless, what riskless arbitrage strategy could a money
The spot price of a widget is $70.00 per unit. Forward prices for 3, 6, 9, and 12 months are $70.70, $71.41, $72.13, and $72.86. Assuming a 5% continuously compounded annual risk-free rate, what are the annualized lease rates for each maturity? Is this an example of contango or backwardation?
Using Table 6.6, what is your best guess about the current price of gold per ounce?
Consider production ratios of 2:1:1, 3:2:1, and 5:3:2 for oil, gasoline, and heating oil. Assume that other costs are the same per gallon of processed oil. a. Which ratio maximizes the per-gallon profit if oil costs $80/barrel, gasoline is $2/gallon, and heating oil is $1.80/gallon? b. Suppose
evening Suppose you know nothing about widgets. You are going to approach a widget merchant to borrow one in order to short-sell it. (That is, you will take physical possession of the widget, sell it, and return a widget at time T.) Before you ring the doorbell, you want to make a judgment about
The current price of oil is $32.00 per barrel. Forward prices for 3, 6, 9, and 12 months are $31.37, $30.75, $30.14, and $29.54. Assuming a 2% continuously compounded annual risk-free rate, what is the annualized lease rate for each maturity? Is this an example of contango or backwardation?
Given a continuously compounded risk-free rate of 3% annually, at what lease rate will forward prices equal the current commodity price? (Recall the copper example in Section 6.3.) If the lease rate were 3.5%, would there be contango or backwardation?
Suppose that copper costs $3.00 today and the continuously compounded lease rate for copper is 5%. The continuously compounded interest rate is 10%. The copper price in 1 year is uncertain and copper can be stored costlessly.a. If you short-sell a pound of copper for 1 year, what payment do you
Suppose the gold spot price is $300/oz, the 1-year forward price is 310.686, and the continuously compounded risk-free rate is 5%.a. What is the lease rate?b. What is the return on a cash-and-carry in which gold is not loaned?c. What is the return on a cash-and-carry in which gold is loaned,
a. What are some possible explanations for the shape of this forward curve?b. What annualized rate of return do you earn on a cash-and-carry entered into in December of Year 0 and closed in March of Year 1? Is your answer sensible?c. What annualized rate of return do you earn on a cash-and-carry
a. Suppose that you want to borrow a widget beginning in December of Year 0 and ending in March of Year 1. What payment will be required to make the transaction fair to both parties?b. Suppose that you want to borrow a widget beginning in December of Year 0 and ending in September of Year 1. What
a. Suppose the March Year 1 forward price were $3.10. Describe two different transactions you could use to undertake arbitrage.b. Suppose the September Year 1 forward price fell to $2.70 and subsequent forward prices fell in such a way that there is no arbitrage from September Year 1 and going
Consider Example 6.1. Suppose the February forward price had been $2.80. What would the arbitrage be? Suppose it had been $2.65. What would the arbitrage be? In each case, specify the transactions and resulting cash flows in both November and February. What are you assuming about the convenience
Suppose you observe the following zero-coupon bond prices per $1 of maturity payment: 0.96154 (1-year), 0.91573 (2-year), 0.87630 (3-year), 0.82270 (4-year), 0.77611 (5-year). For each maturity year compute the zero-coupon bond yields (effective annual and continuously compounded), the par coupon
What is the rate on a synthetic FRA for a 90-day loan commencing on day 90? A 180-day loan commencing on day 90? A 270-day loan commencing on day 90? Discuss.
What is the rate on a synthetic FRA for a 180-day loan commencing on day 180? Suppose you are the counterparty for a borrower who uses the FRA to hedge the interest rate on a $10m loan. What positions in zero-coupon bonds would you use to hedge the risk on the FRA? Discuss.
Suppose you are the counterparty for a lender who enters into an FRA to hedge the lending rate on $10m for a 90-day loan commencing on day 270. What positions in zero-coupon bonds would you use to hedge the risk on the FRA? Discuss.
Using the information in Table 7.1, suppose you buy a 3-year par coupon bond and hold it for 2 years, after which time you sell it. Assume that interest rates are certain not to change and that you reinvest the coupon received in year 1 at the 1-year rate prevailing at the time you receive the
As in the previous problem, consider holding a 3-year bond for 2 years. Nowsuppose that interest rates can change, but that at time 0 the rates in Table 7.1 prevail. What transactions could you undertake using forward rate agreements to guarantee that your 2-year return is 6.5%?In table 7.1
Consider the implied forward rate between year 1 and year 2, based on Table 7.1.a. Suppose that r0(1, 2) = 6.8%. Show how buying the 2-year zero-coupon bond and borrowing at the 1-year rate and implied forward rate of 6.8% would earn you an arbitrage profit.b. Suppose that r0(1, 2) = 7.2%. Show how
Suppose the September Eurodollar futures contract has a price of 96.4. You plan to borrow $50m for 3 months in September at LIBOR, and you intend to use the Eurodollar contract to hedge your borrowing rate.a. What rate can you secure?b. Will you be long or short the Eurodollar contract?c. How many
A lender plans to invest $100m for 150 days, 60 days from today. (That is, if today is day 0, the loan will be initiated on day 60 and will mature on day 210.) The implied forward rate over 150 days, and hence the rate on a 150-day FRA, is 2.5%. The actual interest rate over that period could be
Consider the same facts as the previous problem, only now consider hedging with the 3-month Eurodollar futures. Suppose the Eurodollar futures contract that matures 60 days from today has a price on day 0 of 94.a. What issues arise in using the 3-month Eurodollar contract to hedge a 150-day loan?b.
Consider the bonds in Example 7.8. What hedge ratio would have exactly hedged the portfolio if interest rates had decreased by 25 basis points? Increased by 25 basis points? Repeat assuming a 50-basis-point change. Discuss.
Using the information in the previous problem, find the price of a 5-year coupon bond that has a par payment of $1,000.00 and annual coupon payments of $60.00. Discuss.
Compute Macaulay and modified durations for the following bonds:a. A 5-year bond paying annual coupons of 4.432% and selling at par.b. An 8-year bond paying semiannual coupons with a coupon rate of 8% and a yield of 7%.c. A10-year bond paying annual coupons of 6% with a price of $92 and maturity
Consider the following two bonds which make semiannual coupon payments: a 20 year bond with a 6% coupon and 20% yield, and a 30-year bond with a 6% coupon and a 20% yield.a. For each bond, compute the price value of a basis point.b. For each bond, compute Macaulay duration.c. "For otherwise
An 8-year bond with 6% annual coupons and a 5.004% yield sells for $106.44 with a Macaulay duration of 6.631864. A 9-year bond has 7% annual coupons with a 5.252% yield and sells for $112.29 with a Macaulay duration of 7.098302. You wish to duration-hedge the 8-year bond using a 9-year bond. How
A 6-year bond with a 4% coupon sells for $102.46 with a 3.5384% yield. The conversion factor for the bond is 0.90046. An 8-year bond with 5.5% coupons sells for $113.564 with a conversion factor of 0.9686. (All coupon payments are semiannual.) Which bond is cheaper to deliver given a T-note futures
a. Compute the convexity of a 3-year bond paying annual coupons of 4.5% and selling at par.b. Compute the convexity of a 3-year 4.5% coupon bond that makes semiannual coupon payments and that currently sells at par.c. Is the convexity different in the two cases? Why? Discuss.
Suppose a 10-year zero-coupon bond with a face value of $100 trades at $69.20205.a. What is the yield to maturity and modified duration of the zero-coupon bond?b. Calculate the approximate bond price change for a 50-basis-point increase in the yield, based on the modified duration you calculated
Suppose you observe the following effective annual zero-coupon bond yields: 0.030 (1-year), 0.035 (2-year), 0.040 (3-year), 0.045 (4-year), 0.050 (5-year). For each maturity year compute the zero-coupon bond prices, continuously compounded zero-coupon bond yields, the par coupon rate, and the
Suppose you observe the following 1-year implied forward rates: 0.050000 (1 year), 0.034061 (2-year), 0.036012 (3-year), 0.024092 (4-year), 0.001470 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the
Suppose you observe the following continuously compounded zero-coupon bond yields: 0.06766 (1-year), 0.05827 (2-year), 0.04879 (3-year), 0.04402 (4-year), 0.03922 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual zero-coupon bond yields, the par coupon rate, and
Suppose you observe the following par coupon bond yields: 0.03000 (1-year), 0.03491 (2-year), 0.03974 (3-year), 0.04629 (4-year), 0.05174 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the 1-year
Using the information in Table 7.1,a. Compute the implied forward rate from time 1 to time 3.b. Compute the implied forward price of a par 2-year coupon bond that will be issued at time 1.
Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that will guarantee a 6%effective annual interest rate for 1 year on $500,000.00. On the date you borrow the $500,000.00, the actual interest rate is 5%. Determine the dollar settlement of the FRA assuminga.
Using the same information as the previous problem, suppose the interest rate on the borrowing date is 7.5%. Determine the dollar settlement of the FRA assuminga. Settlement occurs on the date the loan is initiated.b. Settlement occurs on the date the loan is repaid.Use the following zero-coupon
Suppose that 1- and 2-year oil forward prices are $22/barrel and $23/barrel. The 1 and 2-year interest rates are 6% and 6.5%. Show that the new 2-year swap price is $22.483. Discuss.
Using the zero-coupon bond prices and oil forward prices in Table 8.9, what is the price of an 8-period swap for which two barrels of oil are delivered in even-numbered quarters and one barrel of oil in odd-numbered quarters? Discuss.
Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what are gas forward prices for each of the 8 quarters?
Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what is the implicit loan amount in each quarter in an 8-quarter natural gas swap?In table 8.9
What is the fixed rate in a 5-quarter interest rate swap with the first settlement in quarter 2? Discuss.
Using the zero-coupon bond yields in Table 8.9, what is the fixed rate in a 4-quarter interest rate swap? What is the fixed rate in an 8-quarter interest rate swap? Discuss.
What 8-quarter dollar annuity is equivalent to an 8-quarter annuity of = C1?
Using the assumptions in Tables 8.5 and 8.6, verify that equation (8.13) equals 6%. Discuss.
Using the information in Table 8.9, what are the euro-denominated fixed rates for 4- and 8-quarter swaps?
Using the information in Table 8.9, verify that it is possible to derive the 8-quarter dollar interest swap rate from the 8-quarter euro interest swap rate by using equation (8.13). Discuss.
Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22. The 1-year effective annual interest rate is 6.0%, the 2-year interest rate is 6.5%, and the 3-year interest rate is 7.0%.a. What is the 3-year swap price?b. What is the price of a 2-year swap beginning in one
Consider the same 3-year oil swap. Suppose a dealer is paying the fixed price and receiving floating. What position in oil forward contracts will hedge oil price risk in this position? Verify that the present value of the locked-in net cash flows is zero. Discuss.
Consider the 3-year swap in the previous example. Suppose you are the fixed-rate payer in the swap. How much have you overpaid relative to the forward price after the first swap settlement? What is the cumulative overpayment after the second swap settlement? Verify that the cumulative overpayment
Consider the same 3-year swap. Suppose you are a dealer who is paying the fixed oil price and receiving the floating price. Suppose that you enter into the swap and immediately thereafter all interest rates rise 50 basis points (oil forward prices are unchanged). What happens to the value of your
Supposing the effective quarterly interest rate is 1.5%, what are the per-barrel swap prices for 4-quarter and 8-quarter oil swaps? (Use oil forward prices in Table 8.9.) What is the total cost of prepaid 4- and 8-quarter swaps? Discuss.
Using the information about zero-coupon bond prices and oil forward prices in Table 8.9, construct the set of swap prices for oil for 1 through 8 quarters.
Using the information in Table 8.9, what is the swap price of a 4-quarter oil swap with the first settlement occurring in the third quarter? Discuss.
Given an 8-quarter oil swap price of $20.43, construct the implicit loan balance for each quarter over the life of the swap. Discuss.
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