copyright © 2001 by harcourt, inc. all rights reserved.1 chapter 10: futures hedging strategies the...

Copyright © 2001 by Harcourt, Inc. All rights reserved.

1

Chapter 10: Futures Hedging Strategies

The law of the conservation of risk is like the law of the The law of the conservation of risk is like the law of the conservation of misery. You can only pass it around. You conservation of misery. You can only pass it around. You cannot get rid of it.cannot get rid of it.

Tanya Styblo BederTanya Styblo Beder

RiskRisk, February 1999, February 1999


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Important Concepts in Chapter 10

Why firms hedgeWhy firms hedge Hedging conceptsHedging concepts Factors involved when constructing a hedgeFactors involved when constructing a hedge Hedge ratiosHedge ratios Examples of short-term interest rate, intermediate- and Examples of short-term interest rate, intermediate- and

long-term interest rate, and stock index futures hedgeslong-term interest rate, and stock index futures hedges


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Why Hedge? The value of the firm may not be independent of financial decisions The value of the firm may not be independent of financial decisions

becausebecause Shareholders might be unaware of the firm’s risks.Shareholders might be unaware of the firm’s risks. Shareholders might not be able to identify the correct number of Shareholders might not be able to identify the correct number of

futures contracts necessary to hedge.futures contracts necessary to hedge. Shareholders might have higher transaction costs of hedging than Shareholders might have higher transaction costs of hedging than

the firm.the firm. There may be tax advantages to a firm hedging.There may be tax advantages to a firm hedging. Hedging reduces bankruptcy costs.Hedging reduces bankruptcy costs.

Managers may be reducing their own risk.Managers may be reducing their own risk. Hedging may send a positive signal to creditors.Hedging may send a positive signal to creditors. Dealers hedge so as to make a market in derivatives.Dealers hedge so as to make a market in derivatives.


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Why Hedge? (continued) Reasons not to hedgeReasons not to hedge

Hedging can give a misleading impression of the Hedging can give a misleading impression of the amount of risk reducedamount of risk reduced

Hedging eliminates the opportunity to take advantage Hedging eliminates the opportunity to take advantage of favorable market conditionsof favorable market conditions

There is no such thing as a hedge. Any hedge is an act There is no such thing as a hedge. Any hedge is an act of taking a position that an adverse market movement of taking a position that an adverse market movement will occur. This, itself, is a form of speculation.will occur. This, itself, is a form of speculation.


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Hedging Concepts Short Hedge and Long HedgeShort Hedge and Long Hedge

Short (long) hedge implies a short (long) position in futuresShort (long) hedge implies a short (long) position in futures Short hedges can occur becauseShort hedges can occur because

The hedger owns an asset and plans to sell it later.The hedger owns an asset and plans to sell it later. The hedger plans to issue a liability laterThe hedger plans to issue a liability later

Long hedges can occur becauseLong hedges can occur because The hedger plans to purchase an asset later.The hedger plans to purchase an asset later. The hedger may be short an asset.The hedger may be short an asset.

An anticipatory hedge is a hedge of a transaction that is An anticipatory hedge is a hedge of a transaction that is expected to occur in the future. expected to occur in the future.

See See Table 10.1, p. 405Table 10.1, p. 405 for hedging situations. for hedging situations.


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Hedging Concepts (continued) The BasisThe Basis

Basis = spot price - futures price.Basis = spot price - futures price. Hedging and the BasisHedging and the Basis

(short hedge) = ST - S0 (from spot market) - (fT - f0) (from futures market)

(long hedge) = -ST + S0 (from spot market) + (fT - f0) (from futures market)

If hedge is closed prior to expiration,

(short hedge) = St - S0 - (ft - f0)

If hedge is held to expiration, SIf hedge is held to expiration, Stt = S = STT = f = fTT = f = ftt..


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Hedging Concepts (continued) The Basis (continued)The Basis (continued)

Hedging and the Basis (continued)Hedging and the Basis (continued) Example: Buy asset for $100, sell futures for $103. Hold Example: Buy asset for $100, sell futures for $103. Hold

until expiration. Sell asset for $97, close futures at $97. Or until expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure.deliver asset and receive $103. Make $3 for sure.

Basis definitionBasis definition initial basis: binitial basis: b00 = S = S00 - f - f00

basis at time t: bbasis at time t: btt = S = Stt - f - ftt

basis at expiration: bbasis at expiration: bTT = S = STT - f - fTT = 0 = 0

For a position closed at t:For a position closed at t: (short hedge) = St - ff - (S0 - f0) = -b0 + bt


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Hedging Concepts (continued) The Basis (continued)The Basis (continued)

This is the change in the basis and illustrates the illustrates the principle of basis risk.principle of basis risk.

Hedging attempts to lock in the future price of an asset Hedging attempts to lock in the future price of an asset today, which will be ftoday, which will be f00 + (S + (Stt - f - ftt).).

A perfect hedge is practically non-existent.A perfect hedge is practically non-existent. Short hedges benefit from a strengthening basis.Short hedges benefit from a strengthening basis. Everything we have said here reverses for a long hedge.Everything we have said here reverses for a long hedge. See See Table 10.2, p. 408Table 10.2, p. 408 for hedging profitability and the for hedging profitability and the

basis.basis.


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Hedging Concepts (continued)

The Basis (continued)The Basis (continued) Example: March 30. Spot gold $387.15. June futures Example: March 30. Spot gold $387.15. June futures

$388.60. Buy spot, sell futures. Note: b$388.60. Buy spot, sell futures. Note: b00 = 387.15 - = 387.15 -

388.60 = -1.45. If held to expiration, profit should be 388.60 = -1.45. If held to expiration, profit should be change in basis or 1.45.change in basis or 1.45. At expiration, let SAt expiration, let STT = $408.50. Sell gold in spot for = $408.50. Sell gold in spot for

$408.50, a profit of 21.35. Buy back futures at $408.50, a profit of 21.35. Buy back futures at $408.50, a profit of -19.90. Net gain =1.45 or $145 $408.50, a profit of -19.90. Net gain =1.45 or $145 on 100 oz. of gold. on 100 oz. of gold.


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The Basis (continued)The Basis (continued) Example: (continued)Example: (continued)

Instead, close out prior to expiration when SInstead, close out prior to expiration when Stt = =

$377.52 and f$377.52 and ftt = $378.63. Profit on spot = -9.63. = $378.63. Profit on spot = -9.63.

Profit on futures = 9.97. Net gain = .34 or $34 on Profit on futures = 9.97. Net gain = .34 or $34 on 100 oz. Note that change in basis was b100 oz. Note that change in basis was btt - b - b00 or or

-1.11 - (-1.45) = .34.-1.11 - (-1.45) = .34. Behavior of the Basis. See Behavior of the Basis. See Figure 10.1, p. 409Figure 10.1, p. 409..


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Some Risks of HedgingSome Risks of Hedging cross hedgingcross hedging spot and futures prices occasionally move oppositespot and futures prices occasionally move opposite quantity riskquantity risk


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Contract ChoiceContract Choice Which futures commodity?Which futures commodity?

One that is most highly correlated with spotOne that is most highly correlated with spot A contract that is favorably pricedA contract that is favorably priced

Which expiration?Which expiration? The futures whose maturity is closest to but after the The futures whose maturity is closest to but after the

hedge termination date subject to the suggestion not to be hedge termination date subject to the suggestion not to be in the contract in its expiration monthin the contract in its expiration month

See See Table 10.3, p. 412Table 10.3, p. 412 for example of recommended for example of recommended contracts for T-bond hedgecontracts for T-bond hedge

Concept of rolling the hedge forwardConcept of rolling the hedge forward


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Contract Choice (continued)Contract Choice (continued) Long or short?Long or short?

A critical decision! No room for mistakes.A critical decision! No room for mistakes. Three methods to answer the question. See Three methods to answer the question. See Table Table

10.4, p. 41310.4, p. 413

• worst case scenario methodworst case scenario method

• current spot position methodcurrent spot position method

• anticipated future spot transaction methodanticipated future spot transaction method


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Margin Requirements and Marking to MarketMargin Requirements and Marking to Market low margin requirements on futures, butlow margin requirements on futures, but cash will be required for margin callscash will be required for margin calls


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Determination of the Hedge RatioDetermination of the Hedge Ratio Hedge ratio: The number of futures contracts to hedge Hedge ratio: The number of futures contracts to hedge

a particular exposurea particular exposure Naïve hedge ratioNaïve hedge ratio Appropriate hedge ratio should beAppropriate hedge ratio should be

NNff = - = - S/ f Note that this ratio must be estimated.


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Minimum Variance Hedge RatioMinimum Variance Hedge Ratio Profit from short hedge:Profit from short hedge:

= S + fNf

Variance of profit from short hedge:Variance of profit from short hedge:

S2 + f

2Nf2 + 2SfNf

The optimal (variance minimizing) hedge ratio is (see The optimal (variance minimizing) hedge ratio is (see Appendix 10A)Appendix 10A) Nf = - Sf/f

2

This is the beta from a regression of spot price change on futures price change.


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Minimum Variance Hedge Ratio (continued)Minimum Variance Hedge Ratio (continued) Hedging effectiveness is Hedging effectiveness is

• ee** = (risk of unhedged position - risk of hedged = (risk of unhedged position - risk of hedged position)/risk of unhedged positionposition)/risk of unhedged position

• This is coefficient of determination from This is coefficient of determination from regression.regression.


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Price Sensitivity Hedge RatioPrice Sensitivity Hedge Ratio This applies to hedges of interest sensitive securities.This applies to hedges of interest sensitive securities. First we introduce the concept of duration. We start First we introduce the concept of duration. We start

with a bond priced at B:with a bond priced at B:

where CPwhere CPtt is the cash payment at time t and y is the is the cash payment at time t and y is the

yield, or discount rate.yield, or discount rate.

T

1tt

t

y)(1

CPB


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Price Sensitivity Hedge RatioPrice Sensitivity Hedge Ratio An approximation to the change in price for a yield change An approximation to the change in price for a yield change

isis

with DURwith DURBB being the bond’s duration, which is a weighted- being the bond’s duration, which is a weighted-

average of the times to each cash payment date on the bond, average of the times to each cash payment date on the bond, and and represents the change in the bond price or yield. represents the change in the bond price or yield.

Duration has many weaknesses but is widely used as a Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.measure of the sensitivity of a bond’s price to its yield.

y1

y)(DURBB B


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Price Sensitivity Hedge RatioPrice Sensitivity Hedge Ratio The hedge ratio is as follows (See Appendix 10A for The hedge ratio is as follows (See Appendix 10A for

derivation.):derivation.):

Note that DURNote that DURSS -(S/S)(1 + yS)/yS and DURf -(f/f)(1 + yf)/yf

Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.

NDUR

DUR

S

f

1 y

1 yfS

f

f

S


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Price Sensitivity Hedge Ratio (continued)Price Sensitivity Hedge Ratio (continued) Alternatively, Alternatively,

NNff = -(Yield beta)PVBP = -(Yield beta)PVBPSS/PVBP/PVBPff

• where Yield beta is the beta from a regression of where Yield beta is the beta from a regression of spot yields on futures yields and spot yields on futures yields and

• PVBPPVBPSS, PVBP, PVBPff is the present value of a basis is the present value of a basis

point change in the spot and futures prices.point change in the spot and futures prices.


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Stock Index Futures HedgingStock Index Futures Hedging Appropriate hedge ratio isAppropriate hedge ratio is

NNff = - = -(S/f) This is the beta from the CAPM, provided the

futures contract is on the market index proxy. Tailing the HedgeTailing the Hedge

With marking to market, the hedge is not precise With marking to market, the hedge is not precise unless tailing is done. This shortens the hedge ratio.unless tailing is done. This shortens the hedge ratio.

See See Table 10.5, p. 422Table 10.5, p. 422..


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Hedging Strategies

Short-Term Interest Rate HedgesShort-Term Interest Rate Hedges First we need to familiarize ourselves with the basics of T-First we need to familiarize ourselves with the basics of T-

bill and Eurodollar futures.bill and Eurodollar futures. The T-bill futures is priced using the IMM index method.The T-bill futures is priced using the IMM index method.

• Let discount rate be 8.25Let discount rate be 8.25• Futures price is quoted as 100 - 8.25 = 91.75. This is Futures price is quoted as 100 - 8.25 = 91.75. This is

the IMM Index.the IMM Index.• The actual futures price is 100 - 8.25(90/360) = The actual futures price is 100 - 8.25(90/360) =

97.9375 or $979,375 per $1 million contract.97.9375 or $979,375 per $1 million contract.• Each basis point move amounts to $25.Each basis point move amounts to $25.


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Hedging Strategies

Short-Term Interest Rate Hedges (continued)Short-Term Interest Rate Hedges (continued) The Eurodollar futures is also priced using the IMM The Eurodollar futures is also priced using the IMM

index method.index method.• Note that the actual spot Eurodollar pays interest Note that the actual spot Eurodollar pays interest

added on to the principal. For example, if the rate added on to the principal. For example, if the rate is 10%, then $100 deposited for 90 days grows to is 10%, then $100 deposited for 90 days grows to $100(1 + .10(90/360)) = $102.50.$100(1 + .10(90/360)) = $102.50.

• The futures, however, uses the IMM method, as The futures, however, uses the IMM method, as previously illustrated with T-bills, and treats it as previously illustrated with T-bills, and treats it as though the underlying Eurodollar is a discount though the underlying Eurodollar is a discount instrument.instrument.


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Hedging Strategies

Short-Term Interest Rate Hedges (continued)Short-Term Interest Rate Hedges (continued) Hedging the Future Purchase of a Treasury BillHedging the Future Purchase of a Treasury Bill

See See Table 10.6, p. 426Table 10.6, p. 426 for example. for example. Hedging a Future Commercial Paper IssueHedging a Future Commercial Paper Issue

See See Table 10.7, p. 429Table 10.7, p. 429 for example. for example. Hedging a Floating-Rate LoanHedging a Floating-Rate Loan

See See Table 10.8, p. 431Table 10.8, p. 431 for example. for example. This is called a strip hedge. Note also the rolling This is called a strip hedge. Note also the rolling

strip hedge and the stack hedge.strip hedge and the stack hedge.


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Hedging Strategies (continued) Intermediate and Long-Term Interest Rate Futures HedgesIntermediate and Long-Term Interest Rate Futures Hedges

First let us look at the T-note and bond contractsFirst let us look at the T-note and bond contracts T-bonds: must be a T-bond with at least 15 years to T-bonds: must be a T-bond with at least 15 years to

maturity or first call datematurity or first call date T-note: three contracts (2-, 5-, and 10-year)T-note: three contracts (2-, 5-, and 10-year) A bond of any coupon can be delivered but the standard is A bond of any coupon can be delivered but the standard is

a 6% coupon. Adjustments, explained in Chapter 11, are a 6% coupon. Adjustments, explained in Chapter 11, are made to reflect other coupons.made to reflect other coupons.

Price is quoted in units and 32nds, relative to $100 par, Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is 93.4375.e.g., 93 14/32 is 93.4375.

Contract size is $100,000 face value so price is $93,437.50Contract size is $100,000 face value so price is $93,437.50


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Hedging Strategies (continued)

Intermediate and Long-Term Interest Rate Futures Hedges Intermediate and Long-Term Interest Rate Futures Hedges (continued)(continued) Hedging a Long Position in a Government BondHedging a Long Position in a Government Bond

See See Table 10.9, p. 434Table 10.9, p. 434 for example. for example. Anticipatory Hedge of a Future Purchase of a Treasury BillAnticipatory Hedge of a Future Purchase of a Treasury Bill

See See Table 10.10, p. 436Table 10.10, p. 436 for example. for example. Note the use of a regression estimate of the hedge ratio.Note the use of a regression estimate of the hedge ratio.

Hedging a Corporate Bond IssueHedging a Corporate Bond Issue See See Table 10.11, p. 437Table 10.11, p. 437 for example. for example.


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Hedging Strategies (continued)

Stock Index Futures HedgeStock Index Futures Hedge First look at the contractsFirst look at the contracts

We primarily shall use the S&P 500 futures. Its We primarily shall use the S&P 500 futures. Its price is determined by multiplying the quoted price price is determined by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,0001300($250) = $325,000

Stock Portfolio HedgeStock Portfolio Hedge See See Table 10.12, p. 440Table 10.12, p. 440 for example. for example.

Anticipatory Hedge of a TakeoverAnticipatory Hedge of a Takeover See See Table 10.13, p. 442Table 10.13, p. 442 for example. for example.


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Summary Table 10.14, p. 444Table 10.14, p. 444 recaps the types of hedge situations, recaps the types of hedge situations,

the nature of the risk and how to hedge that riskthe nature of the risk and how to hedge that risk


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Appendix 10A: Derivation of the Hedge Ratio Minimum Variance Hedge RatioMinimum Variance Hedge Ratio

The variance of the profit from a hedge isThe variance of the profit from a hedge is

S2 + f

2Nf2 + 2SfNf

Differentiating with respect to Nf, setting to zero and solving for Nf gives

• NNff = - = - SSff//ff22

A check of the second derivative verifies that this is A check of the second derivative verifies that this is a minimum.a minimum.


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Appendix 10A: Derivation of the Hedge Ratio (continued) Price Sensitivity Hedge RatioPrice Sensitivity Hedge Ratio

The value of the position isThe value of the position is V = S + VV = S + VffNNff

Use the following results:Use the following results: VVff//r = r = f/f/rr ys/r = yf/r

Differentiate with respect to r, use the above results, set Differentiate with respect to r, use the above results, set to zero, apply the chain rule and solve for Nto zero, apply the chain rule and solve for Nff. The . The

approximation isapproximation isN

S

f

y

yff

s


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Appendix 10B: Taxation of Hedging

Hedges used by businesses to protect inventory and in Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary standard business transactions are taxed as ordinary income.income.

Transactions must be shown to be legitimate hedges and Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary not just speculation outside of the norm of ordinary business activities. This is called the business motive test.business activities. This is called the business motive test.

copyright © 2001 by harcourt, inc. all rights reserved.1 chapter 10: futures hedging strategies the...

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