*Learning objectives: Calculate the profit and loss on a short or long hedge. Compute the optimal number of futures contracts needed to hedge an exposure and explain and calculate the “tailing the hedge” adjustment. Explain how to use stock index futures contracts to change a stock portfolio’s beta. Explain how to create a long-term hedge using a stack and roll strategy and describe some of the risks that arise from this strategy.*

**Questions:**

21.14.1. Barbara works for an airline. She was been 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, σ(ΔS), is $0.0320. The standard deviation of the daily change in futures prices, σ(ΔF), is $0.0500. The correlation, ρ(ΔS, ΔF), 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:

- The standard deviation of one-day returns in the spot and futures prices, respectively, is σ[r(S)] = 1.30% and σ[r(F)] = 1.50%
- The correlation between the returns, ρ[r(S), r(F)] = 0.90
- The spot and futures price per gallon, respectively, is S(0) = $3.10 and F(0)= $3.60

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)

21.14.2. A tech portfolio with a value of $10.0 million has a beta with respect to the NASDAQ-100, β(P, NASDAQ-100) of 1.20. The desired hedging contract is the E-mini Nasdaq-100 futures contract. The size (i.e., contract unit) of this futures contract is $20.00 * Index Value. If the goal is to reduce the portfolio's beta (from 1.20) to 0.30, how many contracts should be employed?

a. Long 15 contracts

b. Short 5 contracts

c. Short 30 contracts

d. Short 120 contracts

21.14.3. On January 15th of Year 1, a company decides to hedge the planned purchase of 100,000 bushels of corn thirteen months later (on February 15 of Year 2). Below are displayed the per-bushel futures prices of three selected contracts on four different dates.

Assume the following: at the time of ultimate purchase (February 15 of Year 2), the spot price is $6.47 per bushel when the final contract's basis is $6.47 - $6.50 = -$0.03. If the company employs a stack-and-roll strategy, what is the company's net (i.e., after hedging) cost to acquire the corn? [inspired by GARP's EOC Question 8.20]

a. $45,000

b. $602,000

c. $647,000

d. $650,000

**Answers:**

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