Hull.03.06
Category:Hull -> Chapter 03
Questions:
Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. what is the optimal hedge ratio for a 3-month contract? What does it mean?
[my adds]
03.06b If an airline wants to hedge the anticipated purchase of jet fuel and uses oil futures contracts (see here: light sweet crude oil futures) to cross-hedge (i.e., cannot find jet fuel futures), how many contracts are optimal if the purchase will be for 1,000,000 gallons?
03.06c. If the above correlation represents the regression of changes in spot price against changes in futures price, what is the slope of the OLS regression line?
03.06d. If the above correlation represents the regression of changes in spot price against changes in futures price, what is the coefficient of determination (R^2)?
03.06e. What is the covariance between the changes in futures and spot prices? What are the units of this covariance?
03.06f. What is the beta of the change in spot price with respect to the change in futures price?
Answers:
03.06
The optimal hedge ratio is 0.8 x 0.65/0.81 = 0.642
This means that the size of the futures position should be 64.2% of the size of the company’s exposure in a three-month hedge.
03.06b.
A single oil futures contract is for 42,000 gallons. The airline whould short a number of contracts = MV Hedge Ratio * 1 MM gallons / 42,000 gallons per contract = 15.3 or about 15 contracts
03.06c.
0.642. The MV hedge ratio = slope of the regression line = beta of the dependent (explained var) with respect to the independent (explanatory var)
03.06d..
64% = 0.8^2. Note this is not exactly the same as the MV hedge ratio of 0.642, it is just near.
03.06e..
Covariance = Correlation*Volatility*Volatilty. In this case, covariance = 0.4212 dollar^2 = 0.8 correlation * $0.81 * $0.65
Since the correlation is unitless, the units are dollar^2.
03.06f.
0.642 again! The MV hedge ratio = slope of the regression line = beta of the dependent (explained var) with respect to the independent (explanatory var)
Key points to remember:
* Using oil futures to hedge jet fuel is a cross-hedge (implication: basis risk)
* The MV hedge ratio is the result of an OLS regression: change in futures against change in spot
* The slope of the regression line = beta = Cov(f,s) /Var (f) = correlation(f,s)*volatility(f)*volatility(s) / Var(f) = correlation(f,s) * volatility (s) / volatility (f) = correlation (f,s) * “cross-volatility” = minimum variance hedge
* Optimal number of contracts = MV hedge * Position / Size of 1 future contract