1. Chapter 7 Risk Management with Futures Contracts
Be sure to read Sections 4.1 and 4.3 along with this chapter.
Trading futures contracts with the objective of reducing price risk is called hedging.
Not all risks faced by a business can be hedged via a futures market—i.e., quantity risk.

2. Hedging Fundamentals
Hedging with futures typically involves taking a position in a futures market that is opposite the position already held in a cash market.
A Short (or selling) Hedge: Occurs when a firm holds a long cash position and then sells futures contracts for protection against downward price exposure in the cash market.
A Long (or buying) Hedge: Occurs when a firm holds a short cash position and then buys futures contracts for protection against upward price exposure in the cash market. Also known as an anticipatory hedge.
A Cross Hedge: Occurs when the asset underlying the futures contract differs from the product in the cash position
Firms can hold long and short hedges simultaneously (but for different price risks).

3. Profit Profile for a Long Hedge
Change in profit
Q: What’s the “Complaint” about flattening the payoff profile to a horizontal line?
Change in price
A rise in the price of a good will lower profits (or firm value)
A long hedge is appropriate: buy futures to hedge

4. Profit Profile for a Short Hedge
Change in profit
Change in price
A short hedge is appropriate: sell futures to hedge
A decline in the price of a good will lower profits (or firm value)

5. Which Contract Should be Used?
If there is no futures contract on the asset being hedged, use a cross hedge, and select a contract whose price changes are as highly correlated as possible with those of the spot asset.
The liquidity of the contract is also important.
The delivery month should be the same as, or just after, the date the hedge will be lifted.

6. A Strip Hedge vs. a Stacked Hedge
Suppose a firm faces a series of dates (or periods) on which it faces price risk. That is, it has a year (or longer) of production. It can:
Use a “strip” of futures contracts, each with a different delivery date.
Use a stack hedge, in which the most nearby and liquid contract is used, and it is rolled over to the next-to-nearest contract as time passes.

7. Strip Hedge Versus Stacked Hedge: An Example
On March 1, an oil distributor agrees to deliver 10,000 bbl of crude oil in each of the next 8 quarters, at a fixed price. The firm faces the risk that crude oil prices will ___ (rise or fall?), and therefore will enter into a ___ (long or short?) hedge.
On March 1, the firm can:
trade 10 contracts for delivery in each of the next 8 quarters (This process is known as a “strip hedge.”)
trade 80 June contracts. Then, in May, offset the June contracts and trade 70 Sept contracts. Then, in August, offset the Sept contracts and trade 60 Dec contracts, etc. (This process is known as a “stacked hedge.”)
(Note: The last trading day of the June crude oil futures contract is the last business day in May, etc.)

8. Basis and Basis Risk
Basis = Cash price (S) – futures price (F). On the initiation day, basis (S0 – F0) is known.
The basis on the day the hedge is lifted is unknown (a random variable) unless:
The day the hedge is lifted is the contract’s delivery day
The contract’s underlying asset, its quality and its location, are the same as the cash item being hedged.
Otherwise, is a random variable, and the hedger faces “basis risk”.
In a cross hedge, there is always basis risk.

9. The Risk Minimizing Hedge Ratio
Consider the following: DVH = (DS)(QS) – (DF)NFQF, where:
DVH = the value of the ‘hedged portfolio’
QS = the quantity of the spot/cash position being hedged
QF = the number of units of the underlying asset in one futures contract used to hedge (on the opposite side of the cash market position)
NF = the number of futures contracts
DS = change in the spot price of the good
DF = change in the futures price
If DVH = 0, then (DS)(QS) = (DF)NFQF, and the risk-minimizing number of futures contracts to trade, NF*, is
The fractional term, DS/ DF, is the “Hedge Ratio.”

10. Example Using the Hedge Ratio
Suppose you are long 1000 oz. of gold (in the cash market).
There are 100 oz. of gold per futures contract.
For every $0.90 change in the cash market, the futures price changes by $1.00.
You want to engage in a risk minimizing hedge.
What position should you take in the futures market?
How many contracts should you use?

11. Example: Solution
Because you are long in the cash market, using a risk minimizing hedge means that you should take a short position in the futures market. Concerning the number of contracts:

12. Optimal Hedge vs. A Full Hedge
A naïve hedger might think 10 futures contracts should be sold to offset the spot position of long 1000 oz. of gold. This is an example of a “full hedge”, which will not be an optimal hedge when there is basis risk and futures and cash prices don’t move together.
Recall: DVH = (DS)(QS) – (DF)NFQF
Suppose you do the optimal hedge, i.e., long 1000 in spot and short 9 futures: DVH = (90)(1000) – (100)(9)(100) = 0.
Instead, suppose you do a full hedge, i.e., long 1000 in spot and short 10 futures: DVH = (90)(1000) – (100)(10)(100) = -100.

13. Note Bene:
It is really important to note here that we are assuming that the relationship between the changes in the spot price and changes in the futures price will remain the same (i.e., at 0.90 to 1.00) over the time period we are hedging.

14. Hedge Ratio: Example 2
Running the following regression equation results in an estimate of the hedge ratio:
DS = a + b DF + e
Then, b = DS/DF, is an estimate of the hedge ratio.
Suppose you have a long position of 410,000 gallons of heating oil.
You are concerned heating oil prices are going to ___ (rise or fall?), and you want to protect your inventory value. Therefore, you ___ (buy or sell?) heating oil futures).

15. Example 2, Cont.
There are 42,000 gallons of heating oil in one futures contract.
You estimate the following regression equation:
DS = DF
R2 = 0.80
R2 is a goodness of fit measure for the regression model.
It should exceed 0.50 for effective hedging.
The higher it is, the more confident you will be of getting good results.
That is, the higher it is, the more confident you will be that the two prices will move together in the future.

16. Example 2, Cont.
So, sell either 9 or 10 contracts for a risk-minimizing hedge.
Sometimes, hedging is an ‘Art’.

17. The T-Account Approach
You are long 410,000 bbl of heating oil; S0 = $0.74/bbl.
Sell 9 heating oil futures at F0 = $0.78/bbl.
Offset futures at: a)$0.72/bbl, b) $0.741/bbl, c) $0.77/bbl. (Basis risk!)
Sell your oil at $0.70/bbl

18. Another Example Using historical data, you estimate:
You have committed to sell 5000 units of the cash good at the market price one month hence. There are 1000 units of the asset underlying each futures contract.
So: [buy/sell?] (1.31)(5000/1000) = 6.55 contracts.

19. TOTAL REVENUE: $115,000 ($23.00/unit)
Today:
Spot Futures
Commitment to sell Sell 6.55 futures at
5000 units; S0 = F0 = 22.00
Suppose the model works perfectly (S/F = 1.31):
One month hence (scenario 1: ST =24.00; FT =22.763):
Sell good; receive $120,000 Futures Loss = $5,000
TOTAL REVENUE: $115,000 ($23.00/unit)
One month hence (scenario 2: ST = 20.00; FT = ):
Sell good; receive $100, Futures Gain = $15,000

20. Note Bene:
If the futures price and the spot price change in the predicted manner, you lock in the spot price. But in general, you face basis risk. That is
You sell at S2, the spot price in the future, and
You realize profits or losses on the change in F.
With convergence (no basis risk at time T), just sell 5 futures contracts (a full hedge), and you lock in the futures price. Example:

21. TOTAL REVENUE: $110,000 ($22.00/unit)
Today
Spot Futures
Commitment to sell Sell 5 futures at
5000 units; S0 = F0 = 22.00
Suppose we have convergence:
One month hence (scenario 1: ST = 24 = FT )
Sell good; receive $120,000 Loss = $10,000
TOTAL REVENUE: $110,000 ($22.00/unit)
One month hence (scenario 2: ST = 20 = FT)
Sell good; receive $100, Gain = $10,000

22. When Running Your Regression Model,
the R2 is Important. . DS . DS . . . . . DF DF .
Low R2
High R2

23. Dollar Equivalency
Often, you cannot run a regression model to estimate a hedge ratio. In this case, you must estimate the following:
If the spot price changes adversely by, say, a dollar, how much will you lose on your cash position? (= DVS)
If the spot price changes by a dollar, how much of a change do you estimate there will be in the futures price, and hence, in the value of one futures contract? (= DVF)
Then, trade NF* contracts so that
DVS = (NF*)(DVF)

24. Tailing the Hedge, I.
You want to “tail” a hedge when interest rates are high and/or the time until the hedge-lifting date is long.
Instead of trading NF* contracts (as defined previously), trade the present value of NF* contracts:
Where:
d = the number of days until the hedge is anticipated to be lifted.
r = the annualized appropriate interest rate.

25. Tailing the Hedge, II.
As time passes, the present value factor approaches 1.0, and by the day the hedge is lifted, your futures position will equal NF*.
Tailing converts the futures position into a forward position. It negates the effect of daily resettlement, in which profits and losses are realized before the day the hedge is lifted.