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# Continuous Compounding, Duration, Bond future, Swap

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Problem 1
The 6-month, 12-month, 18-month, and 24-month zero rates are 4%, 4.5%, 4.75%, and 5%, with semi-annual compounding.
(a) What are the rates with continuous compounding?
(b) What is the forward rate for the 6-month period beginning in 18 months?
(c) What is the value of an FRA that promises to pay 6% (compounded semi-annually) on a principal of \$1 million for the 6-month period starting in 18 months?

Problem 2
Portfolio A consists of a 1-year zero-coupon bond with a face value of \$2,000 and a 10-year zero-coupon bond with a face value of \$6,000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of \$5,000. The current yield on all bonds is 10% per annum.
(a) Show that both portfolios have the same duration.
(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.
(c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields?

Problem 3
The future price for the June 2005 CBOT bond futures contract is 118-23.
(a) Calculate the conversion factor for a bond maturing on January 1, 2021, paying a coupon of 10%.
(b) Calculate the conversion factor for a bond maturing on October 1, 2026, paying a coupon of 7%.
(c) Suppose that the quoted prices of the bonds in (a) and (b) are 169.00 and 136.00, respectively. Which bond is cheaper to deliver?
(d) Assuming that the cheapest-to-deliver bond is actually delivered, what is the cash price received for the bond?

Problem 4
Suppose that the term structure of interest rates is flat in the United States and Australia. The USD interest rate is 7% per annum and the AUD rate is 9% per annum. The current value of the AUD is 0.62 USD. Under the terms of a swap agreement, a financial institution pays 8% per annum in AUD and receives 4% per annum in USD. The principals in the two currencies are \$12 million USD and 20 million AUD. Payments are exchanged every year, with one exchange having just taken place. The swap will last 2 more years. What is the value of the swap to the financial institution? Assume all interest rates are continuously compounded.

#### Solution Summary

Solution to 4 textbook problem from 'Option, Futures and Other Derivatives, by John C. Hull' dealing with Continuous Compounding, Portfolio Duration, CBOT bond futures contract, Swap.

\$2.19

## What is the arbitrage profit? Which of the following is NOT true? All of the following are types of traders in futures, forward, and options markets EXCEPT: What is the futures price above which there will be a margin call? What hedge ratio should be used when hedging a one month exposure to the price of commodity A? What position should the fund manager take to hedge exposure to the market over the next two months?

Question 1: The price of gold is currently \$700 per ounce. Forward contracts are available to buy or sell gold at \$900 for delivery in one year. An arbitrageur can borrow money at 10% per annum. What is the arbitrage profit?

Question 2: A trader enters into a one-year short forward contract to sell an asset for \$60 when the spot price is \$58. The spot price in one year turns out to be \$63. What is the trader's gain or loss?

Question 3: Which of the following is NOT true?
a. When a CBOE option on IBM is exercised, IBM issues more stock.
b. An American option can be exercised at any time during its life.
c. A call option will always be exercised at maturity if the underlying asset price is greater than the strike price.
d. A put option will always be exercised at maturity if the strike price is greater than the underlying asset price.
e. All of the above are true.

Question 4: Which of the following is NOT true?

a. When a CBOE option on IBM is exercised, IBM issues more stock.
b. An American option can be exercised at any time during its life.
c. A call option will always be exercised at maturity if the underlying asset price is greater than the strike price.
d. A put option will always be exercised at maturity if the strike price is greater than the underlying asset price.
e. All of the above are true.

Question 5: A trader buys 100 European call options (one contract) with a strike price of \$20 and a time to maturity of one year. The cost of each option is \$2. The price of the underlying asset turns out to be \$25 in one year. What is the trader's gain or loss?

Question 6: It is May and a trader writes a September European call option with a strike price of \$20. The stock price is currently \$18 and the option price is \$2. In September, the stock price becomes \$25. What is this trader's cash flow from the option in September?

Question 7: All of the following are types of traders in futures, forward, and options markets EXCEPT:
a. hedgers
b. speculators
c. hackers
d. arbitrageurs
e. All of the above are trader types

Question 8: Which of the following is true?
a. Both forward and futures contracts are traded on exchanges.
b. Forward contracts are traded on exchanges, but futures contracts are not.
c. Futures contracts are traded on exchanges, but forward contracts are not.
d. Neither forward contracts nor futures contracts are traded on exchanges.

Question 9: Which of the following is NOT true?
a. Forward contracts always have zero values.
b. Futures contracts are standardized, forward contracts are not.
c. Delivery or final cash settlement usually takes place with forward contracts; the same is not true for futures contracts.
d. Forward contracts usually have one specified delivery date; futures contracts often have a range of delivery dates.
e. All of the above are true.

Question 10: A company enters into a short futures contract to sell 50,000 pounds of cotton for 70 cents per pound. The initial margin is \$4,000 and the maintenance margin is \$3,000. What is the futures price above which there will be a margin call?

Question 11: Suppose that the standard deviation of monthly changes in the price of commodity A is \$2. The standard deviation of monthly changes in a futures price for a contract on commodity B (which is similar to commodity A) is \$3. The correlation between changes in the futures price and the commodity price is 0.9. What hedge ratio should be used when hedging a one month exposure to the price of commodity A?

Question 12: A fund manager has a portfolio worth \$100 million with a beta of 1.20. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current index level is 850 and one futures contract is on 250 times the index (i.e., the index multiplier is 250). The risk-free rate is 6.0% per annum and the dividend yield on the index is 3.0% per annum. The current three-month futures price is 852.34. What position should the fund manager take to hedge exposure to the market over the next two months?

Question 13: An interest rate is 15% per annum when expressed with quarterly compounding. What is the equivalent rate with continuous compounding?

Question 14: An interest rate is 8% per annum expressed with continuous compounding. What is the equivalent rate with semiannual compounding?

Question 15: The 6-month, 12-month, 18-month, and 24-month zero rates are 3%, 3.5%, 3.75%, and 4% with semi-annual compounding. What is the continuous compounding forward rate for the six-month period beginning in 12 months (i.e., F12,18)?

Question 16: The spot price of an investment asset that provides no income is \$30 and the risk-free rate for all maturities (with continuous compounding) is 10%. What is the three year forward price?

Question 17: The spot price of an investment asset is \$30 and the risk-free rate for all maturities (with continuous compounding) is 10%. The asset provides \$2 income at the end of the first year. What is the three year forward price?

Question 18: The spot price of an investment asset is \$30 and the risk-free rate for all maturities (with continuous compounding) is 10%. The asset provides a 4% continuous yield. What is the three year forward price?

Question 19: An exchange rate is 0.800 and the 2-month domestic risk free interest rate is 5%. The 2-month foreign interest rate is 2% (both rates are continuously compounded). What is the two month forward rate?

Question 20: Which of the following is a consumption asset?
a.The S&P 500 Index
c. Copper
d. IBM Shares
e. None of the above

Question 21: A five-year bond with a yield of 9% (continuously compounded) pays an 8% coupon at the end of each year. What is the duration of the bond?

Question 22: Suppose you enter into an interest rate swap where you are receiving floating and paying fixed. Which of the following is true?
a) Your credit risk is greater when the term structure is upward sloping than when it is downward sloping.
b) Your credit risk is greater when the term structure is downward sloping than when it is upward sloping.
c) Your credit risk exposure increases when interest rates decline unexpectedly.
d) Your credit risk exposure increases when interest rates increase unexpectedly.

Question 23: A trader writes two naked put option contracts. The option price is \$3, the strike price is \$40 and the stock price is \$42. What is the original margin?

Question 24: It is January 15, 2009. The quoted price of a T-bond with a 12% coupon that matures on October 15, 2020, is 102-07. What is the cash price?

Question 25: Suppose Microsoft is paying LIBOR + 1% per annum to its lenders. If Microsoft enters into a interest rate swap with Intel to receive LIBOR and pay fixed 3.9%, what will be the net effect to Microsoft's payment?

Question 26: The current price of a stock is \$94, and three-month European call options with a strike price of \$95 currently sell for \$4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of \$9,400. How high does the stock price have to rise for the option strategy to be more profitable?

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