# Mid-Semester Exam

## Part 1 - Multiple Choice Questions

Questions come from the Hull Test Bank

1. An investor sells a futures contract an asset when the futures price is \$1,500. Each contract is on 100 units of the asset. The contract is closed out when the futures price is \$1,540. Which of the following is true?

1. The investor has made a gain of \$4,000
2. The investor has made a loss of \$4,000
3. The investor has made a gain of \$2,000
4. The investor has made a loss of \$2,000

2. As the convenience yield increases, which of the following is true?

1. The one-year futures price as a percentage of the spot price increases
2. The one-year futures price as a percentage of the spot price stays the same
3. The one-year futures price as a percentage of the spot price decreases
4. Any of the above can happen

3. The time-to-maturity of a Eurodollar futures contract is 4 years and the time-to-maturity of the rate underlying the futures contract is 4.25 years. The standard deviation of the change in the short term interest rate = 0.011. What does the model in the text give as the difference between the futures and the forward interest rate?

1. 0.105%
2. 0.103%
3. 0.098%
4. 0.093%

4. The most recent settlement bond futures price is \$103.5. Which of the following four bonds is the cheapest to deliver?

1. Quoted bond price = 110; conversion factor = 1.0400.
2. Quoted bond price = 160; conversion factor = 1.5200.
3. Quoted bond price = 131; conversion factor = 1.2500.
4. Quoted bond price = 143; conversion factor = 1.3500.

5. Which of the following is closest to the duration of a 2-year bond that pays a coupon of 8% per annum semi-annually? The yield on the bond is 10% per annum with continuous compounding.

1. 1.82
2. 1.85
3. 1.88
4. 1.92

6. Which of the following is true?

1. The optimal hedge ratio is the slope of the best fit line when the spot price (on the y-axis) is regressed against the futures price (on the x-axis).
2. The optimal hedge ratio is the slope of the best fit line when the futures price (on the y-axis) is regressed against the spot price (on the x-axis).
3. The optimal hedge ratio is the slope of the best fit line when the change in the spot price (on the y-axis) is regressed against the change in the futures price (on the x-axis).
4. The optimal hedge ratio is the slope of the best fit line when the change in the futures price (on the y-axis) is regressed against the change in the spot price (on the x-axis).

7. What should a trader do when the one-year forward price of an asset is too low? Assume that the asset provides no income.

1. The trader should borrow the price of the asset, buy one unit of the asset and enter into a short forward contract to sell the asset in one year.
2. The trader should borrow the price of the asset, buy one unit of the asset and enter into a long forward contract to buy the asset in one year.
3. The trader should short the asset, invest the proceeds of the short sale at the risk-free rate, enter into a short forward contract to sell the asset in one year.
4. The trader should short the asset, invest the proceeds of the short sale at the risk-free rate, enter into a long forward contract to buy the asset in one year.

8. The zero curve is upward sloping. Define X as the 1-year par yield, Y as the 1-year zero rate and Z as the forward rate for the period between 1 and 1.5 year. Which of the following is true?

1. X is less than Y which is less than Z
2. Y is less than X which is less than Z
3. X is less than Z which is less than Y
4. Z is less than Y which is less than X

9. The price of a stock on February 1 is \$124. A trader sells 200 put options on the stock with a strike price of \$120 when the option price is \$5. The options are exercised when the stock price is \$110. The trader’s net profit or loss is

Hint: Calculate the profit from the holder’s perspective. Then reverse it for the issuer.

1. Gain of \$1,000
2. Loss of \$2,000
3. Loss of \$2,800
4. Loss of \$1,000

10. A trader buys a call and sells a put with the same strike price and maturity date. What is the position equivalent to?

1. A long forward
2. A short forward
4. None of the above

11. Company X and Company Y have been offered the following rates

 Fixed Rate Floating Rate Company X 3.5% 3-month LIBOR plus 10bp Company Y 4.5% 3-month LIBOR plus 30 bp

Suppose that Company X borrows fixed and company Y borrows floating. If they enter into a swap with each other where the apparent benefits are shared equally, what is company X’s effective borrowing rate?

1. 3-month LIBOR-30 bp
2. 3.1%
3. 3-month LIBOR-10 bp
4. 3.3%

## Part 2 - Essay Questions

1. The one-year zero rate equals 5% and the two-year zero rate equals 5.5%. What is the forward rate for the second year? All rates are continuously compounded.
2. The yield curve is flat at 4% per annum. What is the value of the FRA where the holder pays interest at 5% per annum for a three-month period on a notional principle of \$10,000 starting in three years? All rates are compounded quarterly.
3. Consider a 3×6 FRA on a notional principle amount of \$1 million. The FRA holder borrows at 6% interest rate. The parties settle the FRA after 3 months (90 days) and the settlement is based on the 90-day LIBOR. Assume the 90-day LIBOR equals 8% on the settlement date. All rates are compounded quarterly.

2. The Japanese (Yen) interest rate equals 4% per annum while the Australian dollar (AUD) rate equals 6% per annum. The spot rate is 1 AUD equals 85.62 Yen. In a SWAP agreement, an Australian company pays 5% per annum in Yen and receives 7% per annum in AUD. The principals of the two currencies are 127.5 million Yen and \$1.5 million AUD. Payments are exchange every year with one exchange that the parties completed. The SWAP will last two more years.

3. A one-year gold futures contract is selling for \$1780. Spot gold prices are \$1767 and the one-year risk-free rate is 4%. What arbitrage opportunity is available to investors? What strategy should they use and what will be the profits on the strategy?

4. Suppose that the yield curve is flat at 5% per annum with continuous compounding. A swap with a notional principal of \$100 million in which 6% is received and six-month LIBOR is paid will last another 15 months. Payments are exchanged every six months. The six-month LIBOR rate at the last payment date (three months ago) was 7%.

1. What is the value of the fixed-rate bond underlying the swap?
2. What is the value of the floating-rate bond underlying the swap?
3. What is the value of the swap?

5. A trader wants to pay the fixed rate in a \$20 million FRA for a six month period starting in three years. The three-year LIBOR zero rate is 6% p.a. continuously compounded, and the 3.5 year LIBOR zero rate is 6.5% p.a. continuously compounded.

1. What is the no-arbitrage forward interest rate for the six month period starting in three years, that the trader will agree to pay as the fixed rate in the FRA? Your answer must be in percent per annum, with semi-annual compounding.
2. Six months later, the 2.5 year LIBOR zero rate is 5% p.a. continuously compounded, and the three year LIBOR zero rate is 5.25% p.a. continuously compounded. What is the value of the FRA?

6. Portfolio A consists of a one-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.

1. Show that both portfolios have the same duration.
2. Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same.

7. Explain the role of short put option holder. Under what circumstances will the seller of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.

8. Assume zero rates are 5%, 5.3%, 5.6%, and 6% are the 6-month, 12-month, 18-month, and 24-month respectively with quarterly annual compounding.

1. What are the rates with continuous compounding?
2. What is the two-year par yield? What is the yield on a two-year bond that pays a coupon equal to the par yield?

Multiple Choice

 1. b 2. c 3. b 4. c 5. c 6. c 7. d 8. a 9. d 10. a 11. a

1. a. Convert to continuous rates. F = 6% and applies to payment between 2 and 3 years.
b. Value = -25 and PV = -22.86 or PV = -21.97 for continuous compounding
c. PV = \$4,901.96 or PV = \$4,901.96 continuous compounding

2. The bond discounted cash flows and forward rate methods yield the same answer, which is \$6,569.459 AU

• Answer could have severe rounding error.

3. F = \$1,839.11
Arbitrage

1. You short the gold and receive \$1,767
2. Invest \$1,767 for 4% interest
3. Go long on futures contract for \$1,780

On Maturity

1. Close the bank account and collect \$1,839.11
2. Buy gold using the forward for \$1,780
3. Use gold to close short and earn profit = \$1,839.11 - \$1,780 = \$59.11

PV(payment) = 102.21 million
PV(swap) = 0.396 million

5. FRA = \$20 million (6 / 12)[ x - 0.0972 ]
PV = -0.2667 million

6. a. Portfolio A
PV = Bond price = \$4,016.95
duration = 5.95

Portfolio B
PV = Bond price = \$2,757.81
duration = 5.95

b. The percent change for both portfolios = -0.59%

7. The answer is a plain short put option. Just choose an exercise price, k, and premium, p, and explain it. Then draw a diagram for the profit. Note: profit = p + ST - k

8. a. Interest rates in order 0.0497, 0.0527, 0.0556,, and 0.0596
b. c = 6.0179

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