1. Hedging Strategies Using Futures
Chapter 3

2. Long & Short Hedges: Anticipatory Hedging Rule
Do now in the futures market what you expect to do in the future spot market
A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price
A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

3. Examples of Anticipatory Hedging
Airline goes long gasoline futures to hedge a future purchase of jet fuel.
Firm that will issue 20-year bonds a year from now hedges by shorting T-bond futures.
Farmer shorts wheat futures to hedge his sale of wheat in the future.

4. Opposites Hedging Rule
Your position in the futures market should the opposite of your position in the spot market: if long one, short the other.
A portfolio manager hedges via a short position in stock index futures: spot long, futures short.
Company with outstanding floating-rate debt hedges via long position in T-bill futures: spot short, futures long.

5. Argument in Favor of Corporate Hedging
Companies should focus on the parts of their business in which they possess expertise. They should take steps to minimize risks arising from interest rates, exchange rates, and other market variables as they lack expertise in predicting these variables.

6. Another Argument in Favor of Corporate Hedging
Better (cheaper, more accurate) for company to hedge rather than the individual investors (shareholders) to hedge. The latter do not know the firm’s precise exposure.

7. Arguments against Corporate Hedging
Shareholders are usually well diversified and can make their own hedging decisions; stockholder in an airline also owns share in an oil firm.
Explaining ex-post a situation where there is a loss on the hedge and a gain on the underlying can be difficult, i.e. risk of treasurer being fired.

8. Another Argument against Corporate Hedging
It may increase risk to hedge when competitors do not. Firms in the industry may have the ability to pass on cost increases to customers, i.e. complete pass-thru of cost changes. The variables p (sales price) and c (cost per unit) may be highly positively correlated; a natural hedge exists.
E.g. jewelry manufacturer goes long gold futures. What if gold price subsequently drops?!

9. Examples of natural hedges (complete pass-through of cost)
When p and c are highly positively correlated. Thus, hedging with futures/forward is not warranted.
Gasoline refiner/retailer: retail price vs. crude oil price.
Meat packer (slaughters, processes, distributes meat to retailers): wholesale price vs. live cattle price.

10. Convergence of Futures to Spot (Hedge initiated at time t1 and closed out at time t2)
Price
Spot
Price
Time t1 t2

11. Basis Risk Basis is the difference between spot & futures: B = S - F
Basis risk arises because of the uncertainty about the basis when the hedge is closed out
Hedging involves the substitution of basis risk for spot price risk.

12. Long Hedge Suppose that F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
Hedge via a long futures contract the future purchase of an asset, risk of S2
Cost of Asset=S2 +(F1–F2) = F1 + Basis2

13. Short Hedge Suppose that F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
Hedge via a short futures the future sale of an asset, risk of S2
Price Realized=S2+ (F1 – F2) = F1 + Basis2

14. Choice of Contract
Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge
When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price, aka Cross-hedging. There are then 2 components to the basis.

15. Optimal Hedge Ratio
Proportion of the exposure (a percent) that should optimally be hedged is
where
sS is the standard deviation of DS, the change in the spot price during the hedging period,
sF is the standard deviation of DF, the change in the futures price during the hedging period
r is the coefficient of correlation between DS and DF
Measure of hedging effectiveness is square of r .

16. Analogy: Simple Regression & Optimal Hedge Ratio
Variation to be explained
Risk to be hedged B
Explained variation
Hedged risk C
Unexplained variation
Unhedged risk
R^2 = B/A
% Explained
% Hedged

17. Perfect hedge iff no basis risk
Perfect hedge: R^2 =1. Implies that correlation between S and F = 1.
R^2 is measure of hedging effectiveness. What proportion of variance in spot price is removed by hedging?
No basis risk: variance of (S-F) = 0.
Occurs when the correlation between S and F = 1.

18. Derive number of contracts, N, from h
N = h (QA / QF )
QA is size of exposure
QF is size of futures contract
Example 3.5 p. 60 : Airline wants to hedge purchase 2 months from now of 2M gallons of jet fuel via long position in oil futures contract. Formula, h = .928 (.0263 / ) = .78, i.e. hedge 78% of 2M gallons or 1.56M gallons. How many contracts is that?
Hedging effectiveness=.928^2 or 86%

19. How many contracts, N, is h=78%?
Oil futures contract involves 42,000 gallons i.e., QF= .042M
N = .78 (2M / .042M) = or 37 contracts
Take long position in 37 oil contracts.
Will remove 86% of uncertainty via this hedge.

20. Tailing the Futures Hedge
Adjustment for the fact that the futures hedge generates immediate cash flows (marking to market) whereas the risk being hedged pertain to some time in the future.
NTH= h (VA/VF) = h (QA S/QF F) = N (S/F)
Back to Example 3.5 p. 60 with S = 1.94/gallon F = 1.99/gallon
NTH= (1.94/1.99) = or 36
Effect of tailing the hedge adjustment is to reduce slightly the number of contracts

21. Should you tail the hedge?
Hedge now receipt/payment in the future with a futures contract? Yes!
Why? Futures hedge cash flows start occurring now & continue daily; receipt/payment occurs at future date
Hedge now receipt/payment a month from now with 1-month forward contract? No!
Hedge now receipt/payment a year from now with 1-year forward contract? No!

22. Rolling The Hedge Forward: Hedge a long-term exposure with a time sequence of short-term futures hedges
We can use a series of futures contracts to increase the life of a hedge
Each time we switch from 1 futures contract to another we incur a type of basis risk
Metallgesellschaft debacle: p.69

23. Hedging Using Index Futures
To hedge the risk (reduce to zero the b ) of an investment portfolio the number of contracts that should be shorted is
where P is the value of the portfolio, b is its beta, and F is the current value of one futures (=futures price times contract size)

24. Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.)
Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)

25. Example Futures price of S&P 500 is 1,000
Size of portfolio is $5 million
Beta of portfolio is 1.5
One contract is on $250 times the index
What position in futures contracts on the S&P 500 is necessary to hedge the portfolio? N=1.5(5M/.25M)=30. If short 30 contracts, beta is reduced to zero.

26. More general stock index futures formula
N*= number of contracts that must be held long to change beta of portfolio to beta*
N*= (beta* – beta) (P/F)
Current portfolio exhibits beta
If bullish, may want to raise beta
P=market value of the managed portfolio
F=value of the asset that underlies futures

27. Changing Beta of Managed Portfolio
What position is necessary to reduce the beta of the portfolio to 0.75? N=( )(5M/.25M)=-15; short 15 S&P 500 contracts. What if using Mini S&P 500 contracts, F=0.05M? Short 75 Mini S&P 500 contracts.
What position is necessary to increase the beta of the portfolio to 2.0? N=(2-1.5) (5M/.25M)=10; take a long position in 10 S&P500 contracts.