1. Hedging Strategies Using Futures
Chapter 3

2. Long & Short Hedges
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
A short hedge is also appropriate if you currently own the asset and want to be protected against price fluctuations

3. Arguments in Favor of Hedging
Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

4. Arguments against Hedging
Shareholders are usually well diversified and can make their own hedging decisions
It may increase risk to hedge when competitors do not
Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

5. Convergence of Futures to Spot
Price
Spot Price
Futures
Price
Spot Price
Time
Time
(a)
(b)

6. Basis Risk Basis is the difference between spot & futures
Basis risk arises because of the uncertainty about the basis when the hedge is closed out

7. Long Hedge Suppose that F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future purchase of an asset by entering into a long futures contract
Exposed to basis risk if hedging period does not match maturity date of futures

8. Long Hedge Gain on Futures = F2 - F1 Future Spot Price = S2
Cost of Asset = Future Spot Price - Gain on Futures
Gain on Futures = F2 - F1
Future Spot Price = S2
Cost of Asset= S2 - (F2 - F1)
Cost of Asset = F1 + Basis2
Basis2 = S2 - F2
Future basis is uncertain
Therefore, effective cost of asset hedged is uncertain

9. Short Hedge Suppose that F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future sale of an asset by entering into a short futures contract

10. Short Hedge Gain = F1 - F2 Future Spot Price = S2
Price Realized = Gain on Futures + Future Spot Price
Gain = F1 - F2
Future Spot Price = S2
Price Realized = S2 + F1 - F2
Price Realized = F1 + Basis2
Basis2 = S2 - F2
Price Realized = Cost of Asset
Same Formula

11. Example Hedging period 3 months Futures contract expires in 4 months
We’re exposed to basis risk
Suppose F1 = $105 and S1 =100
Current basis = = -5
Basis in 3 months is uncertain
What is basis in 4 months?

12. Example If F2 = 110 and S2 = $102 Future basis = 102 - 110 = -8
Long Hedge
Gain = = $5
Effective cost = = $97
Short Hedge
Gain = = $ -5
Realized (effective) price = (-5) = $97
Future basis = = -8
Formula: Realized Price = F1 + basis2
Realized Price = (-8) = $97

13. 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.

14. Minimum Variance Hedge Ratio
Hedge ratio is the ratio of the futures to underlying asset position
A perfect hedge requires that futures and underlying spot asset price changes are perfectly correlated
For imperfect hedge, set hedge ratio to minimize variance of hedging error

15. Minimum Variance Hedge Ratio
Proportion of the exposure 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.

16. Minimum Variance Hedge Ratio Continued
Error =S - hF
Perfect hedge: Error = 0 at all times
S = hF + Error
Choose h to minimize var(Error)
Therefore, h = slope of regression
Or h = cov(S, F)/var(F)

17. Hedging Using Index Futures
P: Current portfolio value
A: Current value of futures contract on index
F: Current futures price of index
S: Current value of underlying index
A = F x multiplier
For S&P 500 futures, multiplier = $250
If F=1000, A = 250,000

18. Hedging Using Index Futures
Naive hedge: N = - P/A
Match dollar value of futures to dollar value of portfolio
Suppose you wish to hedge $1M portfolio with S&P 500 hundred futures
Sell 1,000,000/250,000 = 4 contracts

19. Hedging Using Index Futures
Remember total risk can divided up into systematic and unsystematic risk
Systematic risk is measured by the portfolio beta
Hedging with futures allows us to change the systematic risk
But not the unsystematic risk

20. Hedging Using Index Futures
To hedge the risk in a portfolio the number of contracts that should be shorted is
where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract

21. Hedging Stock Portfolios
Hedge reduces systematic -- not unsystematic risk
 is computed with respect to the index underlying the futures contract
Futures should be chosen on underlying index that most closely matches the investor’s portfolio

22. Changing Beta Let * = desired beta Let  = portfolio beta
Then N = (* - )(P/A) (N < 0: sell)
N > 0: buy
If we adopt the above convention, this formula textbook generalizes

23. 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.)

24. Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million
Beta of portfolio is 1.5
What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?
A = 1000x250 = $.25 M
N = (0 – 1.5)(5)/.25 = - 30 contracts (sell)

25. Changing Beta
What position is necessary to reduce the beta of the portfolio to 0.75?
N = (.75 – 1.5)(5)/.25 = -15 contracts (sell)
What position is necessary to increase the beta of the portfolio to 2.0?
N = (2 – 1.5)(5)/.25 = 10 contracts (buy)

26. Rolling The Hedge Forward
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