Answer-first summary for fast verification
Answer: Yes, both of her calculations are correct
The first calculation is correct for a standard hedge: \[ N^* = h^* \times \frac{Q_A}{Q_F} = 0.48 \times \frac{5{,}000{,}000}{42{,}000} \approx 57 \] For a **tailing-the-hedge** adjustment, the optimal hedge ratio uses **returns** rather than price changes, and the number of contracts is adjusted by the spot/futures price ratio: \[ N^* = h^* \times \frac{Q_A}{Q_F} \times \frac{S_0}{F_0} \] Compute the hedge ratio from returns: \[ h^* = \rho \frac{\sigma[r(S)]}{\sigma[r(F)]} = 0.90 \times \frac{1.30\%}{1.50\%} = 0.78 \] Then: \[ N^* = 0.78 \times \frac{5{,}000{,}000}{42{,}000} \times \frac{3.10}{3.60} \approx 80 \] So **both calculations are correct**.
Author: Manit Arora
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Q-21.14.1. Tailing the hedge
Barbara works for an airline. She was asked to calculate the number of futures contracts needed to hedge the airline's exposure to jet fuel. The airline plans to purchase 5.0 million gallons of jet fuel, and the airline will employ heating oil futures contracts. Her first approach suggests 57 futures contracts according to the following assumptions: The standard deviation of the daily change in spot prices, , is `\sigma(\Delta F), is \`0.05`00. The correlation, , is 0.750. Therefore, the optimal hedge ratio is 0.480. Finally, Given each futures contract is on 42,000 gallons, the optimal number of contracts rounds to 57.
Subsequent to this analysis, the board asks her to adjust the estimate for the impact of daily settlement. That is, the board wants to "tail the hedge." Based on the board's request to tail the hedge, Barbara collects the following additional information:
$ and $F(0) = \Barbara then computes an alternative optimal hedge ratio of 0.780 such that under tailing the hedge the optimal number of contracts increases to 80.0. Is she correct, or is there a mistake in her analysis?
A
Yes, both of her calculations are correct
B
Her mistake in the first calculation (where she solved for 57 contracts) was to omit a discount factor
C
In tailing the hedge, her mistake was the switch (from the standard deviation of price changes) to standard deviations of one-day returns
D
In tailing the hedge, her mistake was to switch from quantities (i.e., number of units to hedge and number of units per futures contract) to values (i.e., quantity multiplied by price)
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