# An arbitrage is a strategy where (a) You buy one asset and sell another with a view to make a profit if the one you bought appreciates and the one…

1. An arbitrage is a strategy where

(a) You buy one asset and sell another with a view to make a profit if the one you bought appreciates and the one you sold depreciates.

(b) You construct a series of trades that lead to non-negative cash flows at all points in time and at least one positive cash flow.

(c) You buy and asset with a view of selling it at a higher price.

(d) You buy an overpriced asset and sell an underpriced one.

2. The law of one price states that

(a) Similar but not identical assets will trade in the same market.

(b) Similar but not identical portfolios will have the same risk.

(c) Portfolios with different cashflows cannot have the same price.

(d) Portfolios with identical cashflows will have the same price.

3. A replicating portfolio for a derivative security is

(a) A portfolio consisting of long and short positions in the derivative.

(b) A portfolio that has the same payoffs as the derivative.

(c) A portfolio that has payoffs at least as great as, and in some states greater than, the payoffs from the derivative.

(d) A portfolio that combines the derivative with another derivative on the same underlying so as to make the portfolio riskless.

4. The forward price of an asset that has no holding costs or benefits is equal to

(a) The compounded price of the spot asset, where the compounding takes place at the risk-free rate.

(b) The compounded price of the spot asset, where the compounding takes place at a rate equal to the expected return on the asset.

(c) The market’s expectation of the future spot price of the asset.

(d) Its current spot price.

5. Two assets and have the same spot price today. Asset is expected to grow at 10% over the year and asset is expected to grow at 12%. Which of the following is true if there are no holding costs or benefits for either asset?

(a) Asset ‘s one-year forward price will be less than that of asset .

(b) Asset ‘s one-year forward price will be greater than that of asset .

(c) Asset ‘s one-year forward price will be equal to that of asset .

(d) None of the above can be ascertained with certainty, as it also depends on other factors.

6. The spot price of an asset is $50. The expected return on the asset is 10% a year (in simple terms) and the standard deviation of these returns is 20%. The risk-free rate of interest is 5% a year in simple terms. Assuming no costs or benefits of carry, what is the one-year forward price of the asset?

(a) $52.50

(b) $55.00

(c) $57.50

(d) $60.00

7. Two assets and have the same spot price today. The price of asset is expected to grow at 10% over the next year and that of asset is expected to grow at 10% also. Asset has a standard deviation of returns of 10% over the year and asset has standard deviation of 15%. Which of the following is true if there are no holding costs or benefits?

(a) Asset ‘s one-year forward price will be less than that of asset .

(b) Asset ‘s one-year forward price will be greater than that of asset .

(c) Asset ‘s one-year forward price will be equal to that of asset .

(d) Any of the above may be true.

8. The price of oil is $100 per barrel. Oil prices are expected to grow at 4% a year. The one-year risk-free rate of interest is 2% in simple terms. It costs $1 to store a barrel of oil for one year. If oil has no costs or benefits of carry, what is the theoretical one-year forward price of oil?

(a) $100.00

(b) $102.00

(c) $103.02

(d) $104.00

9. The price of oil is $100 per barrel. Oil prices are expected to grow at 4% a year. The one-year risk-free rate of interest is 2% in simple terms. It costs $1 to store a barrel of oil for one year. If you observe a one-year forward price of oil of $98, what inference could you draw?

(a) The forward price is wrong and an arbitrage is possible.

(b) There may be a benefit of carry in the oil market.

(c) The market is in contango.

(d) Demand for oil is expected to fall in a year.

10. The spot price of gold is $1000 per oz. The one-year risk-free rate is 2% in simple terms. There are no costs or benefits of holding gold. If the one-year forward price of gold is $103, what can you say about the market?

(a) The gold spot market is in disequilibrium.

(b) The demand for gold is expected to rise over the coming year.

(c) You can make arbitrage profits by selling spot and buying forward.

(d) You can make arbitrage profits by selling forward and buying spot.

11. The US dollar-euro spot exchange rate is $1.50/€. If the one-year simple interest rate on dollars is 1% and on euro is 2%, what is the one-year forward rate of dollars per euro?

(a) 1.4748

(b) 1.4853

(c) 1.5000

(d) 1.5149

12. The presence of the delivery option in a futures contract means that

(a) A futures contract may be physically settled but not a forward contract.

(b) All else remaining the same, a futures contract will trade at a lower price than a forward.

(c) All else remaining the same, a futures contract will trade at a higher price than a forward.

(d) A futures contract mat be settled in cash or by delivery of the physical asset.

13. The relationship of forwards and futures is best represented by the following statement(s)

(a) If futures price movements and interest rate movements are positively correlated, then futures prices will be higher than forward prices.

(b) If futures price movements and interest rate movements are negatively correlated, then futures prices will be lower than forward prices.

(c) If futures price movements and interest rate movements are uncorrelated, then futures and forward prices will coincide.

(d) All of the above.

14. Counterparty risk in a futures contract is lower than in a forward contract because

(a) The participants in the futures market are better funded.

(b) The futures contract is marked-to-market on a daily basis.

(c) The futures exchange bears the counterparty risk.

(d) The forward market does not charge commissions that may be used to offset the risk of counterparty failure.

15. A stock has a current price of $20. The risk-free interest rate for a half-year maturity is 6% and the dividend rate is 3%. Assume continuous compounding. What is the six-month forward price of the stock?

(a) $20.30

(b) $20.61

(c) $20.92

(d) $21.24

16. How many years does it take to double your money if the continuously-compounded interest rate is 6%?

(a) 11.55

(b) 12.66

(c) 13.77

(d) 16.66

17. If you wanted to double your money in the same time as in the answer to the previous question, but were using monthly compounding, what would be the rate of interest you would require?

(a) 6.0010%

(b) 6.0068%

(c) 6.0163%

(d) 6.0224%

18. Rolling over short-dated futures contracts is the same as taking one long-dated futures contract if

(a) The interest rates are constant.

(b) The average change in the spot price is zero over the life of the contract.

(c) There is daily mark-to-market of both the short-dated and long-dated contracts.

(d) The size (notional value) of the open position in the futures does not change over the horizon of the transactions.

19. A month ago, the price of an IBM stock was $110 and its volatility was 28%. Today, its price is still $110 but its volatility has gone up to 40%. If the one-month interest rate has not changed over the last month and IBM stock does not pay any dividends (i.e., there are no costs or benefits of carry,) then:

(a) The one-month forward prices of IBM today equals the one-month forward price a month ago

(b) The one-month forward price of IBM today is greater than the one-month forward prices a month ago by $10

(c) The one-month forward prices of IBM today is lower than the one-month forward price a month ago by $10

(d) The one-month forward price of IBM today is $44

20. The spot price of an asset is $50. The expected return on the asset is 10% a year (in simple terms) and the standard deviation of these returns is 20%. The risk-free rate of interest is 5% a year in simple terms. What is the one-year forward price of the asset?

(a) $52.50

(b) $55.00

(c) $60.00

(d) Cannot be calculated from the given information.

21. The replication method identifies the price of a USD/GBP forward rate as a function of

(a) The expected future USD/GBP exchange rate, the GBP interest rates, and the USD interest rates

(b) The spot USD/GBP exchange rate and the volatility of the spot USD/GBP exchange rate

(c) The spot USD/GBP exchange rate, the GBP interest rates, and the USD interest rates

(d) Only the spot USD/GBP exchange rate

22. Consider that the one-year Euro interest rate is greater than the US one-year interest rate. How does the one-year forward exchange rate (USD per EUR) compare to the spot exchange rate (USD per EUR)?

(a) The forward rate is larger than the spot rate

(b) The forward rate is smaller than the spot rate

(c) The forward rate is equal to the spot rate

(d) There is not enough information to answer this question

23. An investor enters into a forward contract to buy 5,000 barrels of oil at $80 a barrel in three months. Two months later, suppose that the one-month forward price of oil is $83 a barrel, and the one-month interest rate is 0%. The value of the contract the investor holds after two months is

(a)

(b) + $15,000

(c) +$400,000

(d) +$415,000

24. An investor enters into a forward contract to purchase 100,000 shares of IBM stock in 2 months at prices of $105 per share. After one month, the investor notes that the forward price for the same contract (which now has a one-month maturity) is $103 per share. She also notes that the one-month discount factor is 0.993. The value of the forward contract held by the investor is

(a)

(b)

(c)

(d)

25. A firm enters into a one-year forward contract to buy refined oil. To hedge itself, the firm simultaneously sells one-year futures contracts on crude oil. In which of the following scenarios might the firm come under cash flows pressure related to these contracts?

(a) Oil prices plummet a day before the maturity of the contracts

(b) Oil prices skyrocket a day before the maturity of the contracts

(c) Oil prices plummet a day after the firm enters the contracts

(d) Oil prices skyrocket a day after the firm enters the contracts

26. “Basis” risk may arise in a hedging situation if

(a) The expiry date of the futures contract and the date on which the hedge is unwound do not coincide.

(b) The futures contract used for hedging relates to a commodity that is somewhat different than that being hedged.

(c) A disconnect between spot and futures markets causes the failure of the convergence of futures to spot at expiry of the futures contract.

(d) All of the above.

Answer d. While (c) is not mentioned explicitly in the text, it evidently causes uncertainty in the cash flows of the hedged position.

27. If changes in spot and futures prices are perfectly correlated over the horizon of a hedge, then

(a) The minimum variance hedge ratio is .

(b) The variance of cash flows from a hedged position under the minimum-variance hedge ratio is zero.

(c) The net cash flow at maturity of the hedge is zero.

(d) The standard deviation of spot price changes must equal the standard deviation of futures price changes.

28. If changes in spot and futures prices are uncorrelated, then

(a) The minimum variance hedge ratio is zero.

(b) Basis risk is zero.

(c) Hedging the spot position with futures is effective because the portfolio is well-diversified.

(d) The minimum variance hedge requires holding equal amounts of spot and futures.

29. If changes in spot and futures prices have a correlation of , then

(a) The hedge ratio is .

(b) The variance of cash flows from a hedged position under the minimum-variance hedge ratio is zero.

(c) The net cash flow at maturity of the hedge is zero.

(d) The standard deviation of spot price changes must equal the negative of the standard deviation of futures price changes.

30. You are hedging a spot position with futures. If the spot asset is more volatile than the corresponding futures, the minimum-variance hedge ratio is

(a) Greater than 1.

(b) Exactly equal to 1.

(c) Less than 1.

(d) Indeterminate, given the information available.

31. You are hedging a spot position with futures. If the spot asset is less volatile than the futures, and there is basis risk, which of the following is surely false:

(a) The minimum-variance hedge ratio is greater than 1.

(b) The minimum-variance hedge ratio is less than or equal to 1.

(c) The minimum-variance hedge ratio is negative.

(d) The minimum-variance hedge ratio is not equal to 1.

32. The correlation between changes in price of a spot asset and futures asset is 90%. The standard deviation of changes in spot prices is $2, and that of futures prices is $2.50. The hedge ratio that minimizes hedge variance is

(a) 0.70

(b) 0.72

(c) 0.80

(d) 0.85

33. The correlation between changes in price of a spot asset and futures asset is 99%. The standard deviation of changes in spot prices is $2, and that of futures prices is $3. What is the standard deviation of a position that is long 5 units of the spot asset and is hedged by shorting 4 units of futures?

(a) 1.5

(b) 2.0

(c) 2.5

(d) 3.0

34. The correlation between changes in price of a spot asset and futures asset is 99%. The standard deviation of changes in spot prices is $2, and that of futures prices is $3. What is the standard deviation of a position that is long 5 units of the spot asset and is optimally (i.e., minimum-variance) hedged by using futures?

(a) 1.41

(b) 1.99

(c) 2.52

(d) 3.11

35. Using a linear regression of changes in spot asset prices on changes in futures asset prices, the minimum-variance hedge ratio may be obtained

(a) As the intercept coefficient in the regression.

(b) As the slope coefficient in the regression.

(c) As the of the regression.

(d) As the square-root of the variance of the residuals from the regression.

36. The covariance of changes between the spot price and futures price is 4. The standard deviation of changes in the futures price is 2. The optimal hedge ratio is

(a) 1

(b) 2

(c) 4

(d) 8

37. If the futures contract used to hedge a spot position is marked-to-market daily, then the minimum-variance hedge ratio formula computed ignoring daily resettlement is, in absolute terms,

(a) Biased downwards.

(b) Unbiased.

(c) Biased upwards.

(d) Biased downwards only if interest rates are nonzero.

38. You own an equity portfolio that has a value of $10,000 and a beta of 1.2. The futures price per contract is currently $1,000. How many futures contracts do you need to sell to bring your equity portfolio’s beta to a value of 1?

(a) 0.5

(b) 1.0

(c) 1.5

(d) 2.0

39. The tailed hedge ratio (which takes into account daily resettlement of the futures contract) is smaller than the untailed one in absolute value. Which of these statements is true in relation to this mathematical fact?

(a) The interest earned or lost on the daily mark-to-market gains and losses increases the volatility of the changes in value of the hedging futures position, thereby reducing the hedge ratio.

(b) The volatility of interest rates makes the correlation of spot and futures lower, and enhances basis risk between the spot and futures markets.

(c) If nominal interest rates were constant, the tailed and untailed hedge ratios would be the same.

(d) If real interest rates were constant, the tailed and untailed hedge ratios would be the same.

40. The change in spot prices has a standard deviation of $1. The change in futures prices has a standard deviation of $1.25. The correlation of spot and futures prices is 1. If the daily risk free interest rate is (corresponding to a continuously-compounded rate of 2% per year), then what is the tailed hedge ratio for a spot position hedged by a 30-day futures contract?

(a) 0.7889

(b) 0.7922

(c) 0.7994

(d) 0.8000

41. What must be the daily interest rate (expressed in continuously-compounded and annualized terms) for the tailed hedge ratio to be 90% of the untailed one for a one-year hedge? Assume a hedging horizon of 365 days.

(a) 10%

(b) 15%

(c) 20%

(d) 25%

42. The tailed hedge ratio becomes lower in comparison to the untailed one when

(a) Interest rates rise and hedge maturity increases.

(b) Interest rates rise and hedge maturity decreases.

(c) Interest rates fall and hedge maturity increases.

(d) Interest rates fall and hedge maturity decreases.

43. Suppose you want to hedge a futures contract A with another futures contract B. You calculate the minimum-variance hedge ratio ignoring daily resettlement (for example, by regressing daily changes in Contract A’s prices on daily changes in Contract B’s prices). Suppose, however, that both contracts are marked-to-market daily. Which of the following statements is always true?

(a) The tailed hedge ratio is lower than the untailed one.

(b) The tailed hedge ratio is equal to the untailed one.

(c) The tailed hedge ratio is greater than the untailed one.

(d) None of the above is always true.

44. When the correlation between two assets is exactly , which of the following statements is true?

(a) The hedge ratio is positive.

(b) The hedge ratio is zero.

(c) The variance of the hedged position is maximized.

(d) There is no basis risk in hedging.

45. The tailed minimum-variance hedge ratio becomes lower in comparison to the untailed one when

(a) Nominal interest rates rise and hedge maturity increases.

(b) Real interest rates rise and hedge maturity decreases.

(c) Nominal Interest rates fall and hedge maturity increases.

(d) Real interest rates fall and hedge maturity decreases.

46. If the minimum-variance hedge ratio is +1, then which of the following is true?

(a) There is no basis risk.

(b) The variance of the hedged cash flows is zero.

(c) Both (a) and (b) must be true.

(d) Neither (a) nor (b) is necessarily true.

47. If the minimum-variance hedge ratio is , then which of the following statements is true?

(a) Changes in spot and futures prices are perfectly negatively correlated.

(b) The standard deviations of spot and futures price changes are the same.

(c) The minimum-variance hedge for a long spot exposure is a short futures exposure of the same size.

(d) All of the above.

48. A US-based corporation has decided to make an investment in Norwegian Kroner of NOK 500 Million (NOK = Norwegian Kroner) in 3 months. The company wishes to hedge changes in the the US dollar-NOK exchange rate using forward contracts on either the euro (EUR) or the Swiss Franc (CHF). The company makes the following estimates:

•If EUR forwards are used: The standard deviation of quarterly changes in the USD/NOK spot exchange rate is 0.005, the standard deviation of quarterly changes in the USD/EUR forward rate is 0.025, and the correlation between the changes is 0.90.

•If CHF forwards are used: The standard deviation of quarterly changes in the USD/NOK spot exchange rate is 0.005, the standard deviation of quarterly changes in the USD/CHF forward rate is 0.020, and the correlation between the changes is 0.80.

The current USD/NOK spot rate is 0.160 (i.e., USD 0.160 per NOK), the current 3-month USD/EUR forward rate is 1.36, and the current 3-month USD/CHF forward rate is 1.04.

If the company wishes to carry out a minimum-variance hedge, which currency should it use for this purpose?

(a) CHF, because the forward exchange rate of near 1 makes it “more” like the USD.

(b) EUR, because Norwegian trade with the eurozone countries far exceeds its trade with Switzerland.

(c) EUR, because of the higher correlation of 0.90.

(d) CHF, because of the lower correlation of 0.80.

49. Refer again to the data in Question 48. The minimum-variance hedge, if EUR were to be used for the hedge, is a forward contract calling for the delivery of

(a) EUR 500 million.

(b) EUR 90 million.

(c) EUR 122.4 million.

(d) EUR 367.65 million.

50. Refer again to the data in Question 48. The minimum-variance hedge, if CHF were to be used for the hedge, is a forward contract calling for the delivery of

(a) CHF 500 million.

(b) CHF 100 million.

(c) CHF 104 million.

(d) CHF 96.2 million.

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