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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

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price

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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

• 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

• 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

• 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

where

S is the standard deviation of S, the change in the spot price during the hedging period,

F is the standard deviation of F, the change in the futures price during the hedging period

is the coefficient of correlation between S and F.

h S

F

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• Error =S - hF

• Perfect hedge: Error = 0 at all timesS = hF + Error

• Choose h to minimize var(Error)

• Therefore, h = slope of regression

• Or h = cov(S, F)/var(F)

Minimum Variance Hedge Ratio Continued

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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

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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

• where P is the value of the portfolio, is its beta, and A is the value of the assets underlying one futures contract

P

A

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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

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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

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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