Question 1

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.

Accepted Answer

depends also on the correlation between the changes in spot and futures,and no information is provided on this quantity.

Question 2

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.

Accepted Answer

The tailed hedge ratio accounts for the time value of money, and when interest rates rise, the present value of future cash flows decreases, leading to a lower tailed hedge ratio. Additionally, as hedge maturity increases, the effect of discounting becomes more pronounced, further lowering the tailed hedge ratio compared to the untailed one.

Question 3

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

Accepted Answer

To adjust the portfolio's beta to 1, you need to calculate the number of futures contracts to sell using the formula: $ \text{Number of contracts} = \frac{\beta_{target} - \beta_{portfolio}}{\beta_{futures}} \times \frac{\text{Value of Portfolio}}{\text{Futures Price per Contract}} $. Here, the target beta ($ \beta_{target} $) is 1, the portfolio's current beta ($ \beta_{portfolio} $) is 1.2, and assuming the futures have a beta of 1 ($ \beta_{futures} = 1 $), the calculation is $ \frac{1 - 1.2}{1} \times \frac{10,000}{1,000} = -0.2 \times 10 = -2 $. Selling 2 futures contracts would adjust the portfolio's beta to 1. Negative sign indicates selling futures.

Question 4

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.

Accepted Answer

CThe minimum-variance hedge ratio is calculated based on the volatility of the spot asset relative to the futures. If the spot asset is less volatile than the futures, the hedge ratio (which is the ratio of the futures price change to the spot price change) would not exceed 1, making A surely false. A negative hedge ratio (C) implies that the spot and futures prices move in opposite directions, which is not necessarily true in this context.

Question 5

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%

Accepted Answer

The tailed hedge ratio being 90% of the untailed one implies a reduction due to the effect of daily compounding interest over a year. The formula relating the tailed and untailed hedge ratios involves the exponential function of the continuously compounded rate times the hedging horizon. To achieve a 10% reduction (i.e., to 90% of the original), a specific rate must be chosen that, when applied continuously over 365 days, results in this exact reduction. A 20% annualized continuously compounded rate is the one that fits these criteria, as it provides the necessary exponential decay factor over a year to reduce the hedge ratio to 90% of its original value.

Question 6

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 $R = 1.000055$ (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

Accepted Answer

The answer of The change in spot prices has a...

Question 7

When the correlation between two assets is exactly $- 1$ ,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.

Accepted Answer

When the correlation between two assets is exactly -1, it means they move perfectly inversely to each other. This perfect negative correlation implies that hedging one asset with the other can eliminate all basis risk, as any loss in one would be perfectly offset by a gain in the other.

Question 8

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.

Accepted Answer

The higher correlation should be the determining factor as explained in the chapter.

Question 9

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.

Accepted Answer

presumes negative interest rates,which even if they occurred does not guarantee that the tailed hedge ratio is lower than the untailed one.

Question 10

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

Accepted Answer

Use the formula provided in the text.The calculations are as follows:
> rho = 0.99
> sigDS = 2
> sigDF = 3
> n=5
> m=4
> vr = n^2*sigDS^2 + m^2*sigDF^2 - 2*rho*m*n*sigDS*sigDF
> vr
[1] 6.4
> sqrt(vr)
[1] 2.529822

Question 11

If the minimum-variance hedge ratio is $- 1$ ,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.

Accepted Answer

The answer of If the minimum-variance hedge ratio is \(-...

Question 12

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.

Accepted Answer

The carry on the MTM value cancels out,and there is no tail adjustment factor.

Question 13

If the futures contract used to hedge a spot position is marked-to-market daily,then the minimum-variance hedge ratio formula $h ^ { * } = \rho \times \sigma ( \Delta S ) / \sigma ( \Delta F )$ 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.

Accepted Answer

The tailed hedge ratio (denoted $h ^ { ** } i$n the text)is obtained by multiplying the untailed hedge ratio by a factor that is less than one.

Question 14

If changes in spot and futures prices are perfectly correlated over the horizon of a hedge,then
A)The minimum variance hedge ratio is $+ 1$
)
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.

Accepted Answer

The answer of If changes in spot and futures prices...

Question 15

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

Accepted Answer

The optimal hedge ratio (h*) is calculated using the formula: h* = ρ(σ_S/σ_F), where ρ is the correlation between the spot and futures prices, σ_S is the standard deviation of the spot price, and σ_F is the standard deviation of the futures price. Substituting the given values: h* = 0.99 * (2/3) = 0.66. The variance of the hedged position (σ^2_hedged) is given by: σ^2_hedged = (σ_S^2 + (h*^2 * σ_F^2) - 2h*ρσ_Sσ_F). Substituting the values: σ^2_hedged = (2^2 + (0.66^2 * 3^2) - 2*0.66*0.99*2*3) = 2. The standard deviation of the hedged position is the square root of the variance, √2 ≈ 1.41.

Question 16

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.

Accepted Answer

If changes in spot and futures prices are uncorrelated, there is no relationship between the two variables, and as such, there is no minimum variance hedge ratio. In other words, holding a futures position would not reduce the variance of the spot position, so there is no need to hedge.

Question 17

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.

Accepted Answer

The tail factor depends only on nominal interest rates and maturity.

Question 18

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 $R ^ { 2 }$
Of the regression.
D)As the square-root of the variance of the residuals from the regression.

Accepted Answer

The minimum-variance hedge ratio is obtained as the slope coefficient in the regression of changes in spot asset prices on changes in futures asset prices. This slope, often denoted as beta ($\beta$), indicates how much the spot price changes in response to a change in the futures price, which is crucial for determining the optimal hedge ratio.

Question 19

If changes in spot and futures prices have a correlation of $- 1$ ,then
A)The hedge ratio is $- 1$
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.

Accepted Answer

The answer of If changes in spot and futures prices...

Question 20

"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.

Accepted Answer

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

Quiz 5: Hedging With Futures Forwards

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

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

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?

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:

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.

When the correlation between two assets is exactly $- 1$ ,which of the following statements is true?

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?

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?

If the minimum-variance hedge ratio is $- 1$ ,then which of the following statements is true?

If the futures contract used to hedge a spot position is marked-to-market daily,then the minimum-variance hedge ratio formula $h ^ { * } = \rho \times \sigma ( \Delta S ) / \sigma ( \Delta F )$ computed ignoring daily resettlement is,in absolute terms,

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

If changes in spot and futures prices are uncorrelated,then

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

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

If changes in spot and futures prices have a correlation of $- 1$ ,then

"Basis" risk may arise in a hedging situation if