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Hedging Strategies
Using Futures
Chapter 3
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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
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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
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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
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Convergence of Futures to Spot
Futures
Price
Spot Price
Futures
Price
Spot Price
Time
(a)
Time
(b)
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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
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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
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Long Hedge
• Cost of Asset = Future Spot Price - Gain on Futures
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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
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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
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Short Hedge
• Price Realized = Gain on Futures + Future Spot Price
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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
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Example
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Hedging period 3 months
Futures contract expires in 4 months
We’re exposed to basis risk
Suppose F1 = $105 and S1 =100
Current basis = 100 - 105 = -5
Basis in 3 months is uncertain
What is basis in 4 months?
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Example
• If F2 = 110 and S2 = $102
• Long Hedge
– Gain = 110 - 105 = $5
– Effective cost = 102 - 5 = $97
• Short Hedge
– Gain = 105 - 110 = $ -5
– Realized (effective) price = 102 + (-5) = $97
• Future basis = 102 - 110 = -8
• Formula: Realized Price = F1 + basis2
• Realized Price = 105 + (-8) = $97
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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.
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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
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Minimum Variance Hedge Ratio
Proportion of the exposure that should optimally be
hedged is
sS
hr
sF
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.
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Minimum Variance Hedge
Ratio Continued
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Error =DS - hDF
Perfect hedge: Error = 0 at all times
DS = hDF + Error
Choose h to minimize var(Error)
Therefore, h = slope of regression
Or h = cov(DS, DF)/var(DF)
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Hedging Using Index Futures
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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
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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
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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
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Hedging Using Index Futures
• To hedge the risk in a portfolio the
number of contracts that should be
shorted is
P
b
A
• where P is the value of the portfolio, b is
its beta, and A is the value of the assets
underlying one futures contract
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Hedging Stock Portfolios
• Hedge reduces systematic -- not
unsystematic risk
• b 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
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Changing Beta
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Let b* = desired beta
Let b = portfolio beta
Then N = (b* - b)(P/A) (N < 0: sell)
N > 0: buy
If we adopt the above convention, this
formula textbook generalizes
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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.)
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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)
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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)
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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