1. Hedge Ratio (HR):
The hedge ratio is the ratio of the size of the futures position relative to the spot position. It represents how many futures contracts are needed to hedge against the price movements in the spot position effectively.
HR = (Correlation between spot and futures) x (Standard deviation of spot) / (Standard deviation of futures)
Given values:
Correlation between spot and futures (ρ) = 0.925
Standard deviation of spot (σ_spot) = $0.10
Standard deviation of futures (σ_futures) = $0.125
Now, plug in the values to calculate the hedge ratio:
HR = 0.925 x (0.10) / (0.125) = 0.74
So, the hedge ratio is 0.74. This means that for every $1 worth of spot position, the trader should take a futures position of $0.74 to optimize the hedge.
2. Coefficient of Determination (R²):
The coefficient of determination (R²) measures the goodness-of-fit of a regression. In this case, we are considering a simple linear regression, where the independent variable is the change in futures prices, and the dependent variable is the change in spot prices.
R² = (Correlation between spot and futures)²
Given value:
Correlation between spot and futures (ρ) = 0.925
Now, plug in the value to calculate the coefficient of determination (R²):
R² = 0.925² ≈ 0.855
The R² value is approximately 0.855. It indicates that around 85.5% of the variance in the spot prices can be explained by the variance in the futures prices.
Example:
Suppose the spot price of the currency is currently $100 per unit. The trader holds $100,000 worth of spot position (i.e., 100,000 / 100 = 1000 units). To hedge this spot position optimally, the trader should take a futures position of 0.74 * 1000 = $740. So, the trader would need to buy or sell futures contracts worth $740 to effectively hedge against the price movements in the spot position.
If the correlation between spot and futures is high (as in this case, 0.925), it indicates that the two prices move together closely. Consequently, the hedge will be more effective in reducing the variance of the combined hedge position.
The R² value of approximately 0.855 means that 85.5% of the variability in spot prices can be explained by the variability in futures prices. A higher R² indicates a better fit of the regression line, indicating a stronger relationship between spot and futures prices. This, in turn, enhances the effectiveness of the hedge.
Sure! Here are some multiple-choice questions based on the given information:
1. What is the optimal hedge ratio (HR) for the currency trader?
a) 0.925
b) 0.74
c) 0.100
d) 0.125
Answer: b) 0.74
2. The hedge ratio (HR) is calculated as the:
a) Correlation between spot and futures
b) Standard deviation of spot prices
c) Standard deviation of futures prices
d) Ratio of the size of futures to spot positions
Answer: a) Correlation between spot and futures
3. The coefficient of determination (R²) measures:
a) The optimal hedge ratio (HR)
b) The effectiveness of the hedge
c) The goodness-of-fit of a regression
d) The standard deviation of spot prices
Answer: c) The goodness-of-fit of a regression
4. If the correlation between spot and futures is 0.925, the coefficient of determination (R²) will be approximately:
a) 0.925
b) 0.74
c) 0.855
d) 0.100
Answer: c) 0.855
5. If the annual standard deviation of the spot price is $0.10 and the hedge ratio is 0.74, what would be the size of the futures position to optimally hedge a spot position of $50,000?
a) $50,000
b) $37,000
c) $37,500
d) $27,000
Answer: b) $37,000
Note: For question 5, you can calculate the futures position by multiplying the spot position by the hedge ratio (0.74 * $50,000 = $37,000).
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