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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**

Price Spot Price Futures Price Spot Price Time Time (a) (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 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

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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 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

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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 = = -5 Basis in 3 months is uncertain What is basis in 4 months?

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**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

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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 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**

Error =S - hF Perfect hedge: Error = 0 at all times S = hF + Error Choose h to minimize var(Error) Therefore, h = slope of regression Or h = cov(S, F)/var(F)

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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, 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 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

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Hedging Strategies Using Futures Chapter 3 1 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008.

Hedging Strategies Using Futures Chapter 3 1 Options, Futures, and Other Derivatives, 7th Edition, Copyright © John C. Hull 2008.

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