Lecture 3 Moodle

8/10/2019 Lecture 3 Moodle

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Lecture 3. Hedging strategies using futures

1. Long hedge and short hedge2. An important concept: basis risk

3. Optimal hedge ratio4. Hedging an equity portfolio5. Mini quiz

ReadingHull: Chapter 3


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Long hedge and short hedge

Long Hedge:

involves taking a long position in a futures contract appropriate when the hedger knows it will have to purchase a

certain asset in the future Example

Short Hedge: involves taking a short position in a futures contract appropriate when the hedger owns an asset and expects to sell it at

some time in the future. example

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Example of a long hedge with commodity futures

A producer of electricity cables requires 50 tonnes of copper in

3 months time to meet a certain contract. The spot price ofcopper at the LME is $7724 per tonne, and the futures price for3 months delivery is $7695 per tonne. The size of the contract is25 tonnes.

The producer can hedge its position by :1. taking a long position (a long hedge) in 2 futures contracts

today2. closing out the futures position in 3 months time

3. buying 50 tonnes of copper at the spot market in 3 months


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Suppose the spot price of copper in 3 months time is $7710 per tonne. ( Note: At maturity, the spot price equals thefutur es price. FT=ST=$7710 .) How much does producer gain/lose on the futurescontracts?

50*($7710- $7695) = $750, which is a gain,

What is the total cost of the strategy?

It pays $7710*50 = $385,500 for 50 tonnes ofcopper in the spot market,

making the total cost to be $385,500 - $750 =$384,750

Payoff in the futuresmarket: random

Cost in the spotmarket: random

Hedged cost: certain


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For an alternative outcome, assume the spot price in 3

months time is $7600 per tonne. What is the gain/loss on the futures contracts?

50 * ($7600 - $7695) = - $4750, which is a loss

What is the total cost of the strategy?

It pays $7600*50 = $380,000,

making the total cost $380,000 + $4750 =$384,750 .

Payoff in the futures

market: random

Cost in the spotmarket: random

Hedged cost: certain


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Basis risk Basis is usually defined as the spot price minus the futures price

Basis = Spot price - Futures price At maturity basis equals zero. Before expiration, the difference between the spot and the futures

changes randomly Basis increasing is referred to as the basis strengthening ; Basis decreasing is referred to as the basis weakening ;

Basis risk arises because of the uncertainty about the basis when the hedgeis closed out

The asset whose price is to be hedged is not the same as the assetunderlying the futures contract ( cross hedging ) The maturity time of the futures contract used for hedging does not

match the hedging horizon.7


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Basis risk for long hedgersDefine

F 1 : Futures price when hedge is set up F 2 : Futures price when the asset is purchasedS 2 : Spot price at the time of purchaseb2 : Basis at the time of purchase= S 2 F 2

So if the basis strengthens, the long hedgers positionworsens; if the basis weakens, the long hedgers positionimproves. 8

Cost of asset in the spot market S 2

Gain on Futures F 2 F 1

Effective amount paid for the asset S 2 ( F 2 F 1) = F 1 + b2


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Basis risk for short hedgers

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Define F 1 : Futures price at time hedge is set up F 2 : Futures price at time asset is soldS 2 : Spot price at time of saleb2 : Basis at time of sale =S 2 F 2

So if the basis strengthens, the short hedgers positionimproves; if the basis weakens, the short hedgers positionworsens.

Price of asset received in the spot market S 2

Gain on Futures F 1 F 2

Effective amount received S 2 + ( F 1 F 2) = F 1 + b2


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A numerical example

It is March 1. A US company expects to receive 50 million Japaneseyen at the end of July. Yen futures contracts on the CME Group havedelivery months of March, June, September, and December. Onecontract is for the delivery of 12.5million yen. The company thereforetook a short position in four September yen futures contracts onMarch 1. When the yen are received at the end of July, the companycloses out its position. We suppose that the futures price on March 1in cents per yen is 0.7800 and that the spot and futures prices whenthe contract is closed out are 0.7200 and 0.7250, respectively. What isthe effective price obtained in cents per yen?

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Answer The gain on the futures contract is

F 1 F 2 = 0.7800-0.7250=0.0550 The effective price obtained in cents per yen is the final

spot price plus the gain on the futures:

S 2 + ( F 1 F 2) =0.7200+0.0550=0.7750 a 2nd way of calculating the effective price:

The basis when the contract is closed out is: b= S 2 F 2 =0.7200-0.7250= -0.0050

The effective price is also equal to the futures price when thecontract is entered into adjusted by the basis:

S 2 + ( F 1 F 2) = F 1 + b2=0.7800+(-0.0050)=0.775011


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Choice of Contract

Choice of the delivery month as close as possible to, but later than, the end of the life of the

hedge.1) low basis risk2) avoid erratic futures prices3) eliminate the risk of having to take delivery

Choice of the underlying asset: Cross hedging: Hedging an exposure to the price of one asset with

a contract on another asset. the underlying asset of the chosen futures contract is most closely

correlated with price of the asset being hedged.

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Example

There is no futures contract on jet fuel. An airline company has to buy3 million gallons of jet fuel in three months. Suppose you are incharge of this companys hedging activity. You gather the followingdata on the correlations between jet fuel cash price changes and somenear month energy futures prices:

Which energy futures contract will you choose for hedging jet fuel purchase? Briefly explain.

Is it a long hedge or a short hedge? Briefly explain.13

Correlations

Heating oil 0.54

Gasoline 0.41

Crude oil 0.45


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Optimal Hedge Ratio

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Hedge ratio: the ratio of the size of a position in a hedginginstrument to the size of the position being hedged.

The optimal hedge ratio is :

s S is the standard deviation of DS , the change in thespot price during the hedging period,

s F is the standard deviation of D F , the change in thefutures price during the hedging period

r is the coefficient of correlation between DS and D F .

F

S hs

sr

*


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Derivation of the optimal hedge ratio the change in the value of the portfolio is

S- hF when the hedge is long asset + short futures contracthF-S when the hedge is short asset + long futures contract

In both cases the variance of the hedged portfolio is:

Optimal hedge ratio is the value of h that minimizes thevariance of the hedging portfolio.

The first order condition with respect to h:

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2 2 2 2 2 s F s F h hs s s rs s

F

S

F S F

h

hh

s s

r

s rs s s

022 2

2


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Optimal Number of Contracts

Q A Size of position being hedged (units)

Q F Size of one futures contract (units)

V A Value of position being hedged (=spot price time Q A )

V F Value of one futures contract (=futures price times Q F )

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Optimal number of contracts ifno tailing adjustment

F

A

QQh *

Optimal number of contractsafter tailing adjustment toallow for daily settlement of

futures

F

A

V V h *


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Example question

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An airline needs to purchase 2 million gallons of jetfuel in three months and hedges using heating oilfutures. The spot price is $1.94/gallon and the

futures price is $1.99/gallon. The size of one heatingoil futures contract is 42,000 gallons. From historicaldata, the airline calculated the following: F =0.0313,S =0.0263, and r= 0.928 . Please help the airline to

calculate the optimal hedge ratio, the optimal numberof contracts with and without tailing adjustment.


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Example continued The optimal hedge ratio:

Optimal number of contracts assuming no daily settlement

Optimal number of contracts after tailing

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03.37000,42

000,000,27777.0*

F

A

QQ

h

7777.00313.0

0263.0928.0*

F

S hs s

r

10.3699.1000,42

94.1000,000,27777.0*

F

A

V V

h


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Hedging an equity portfolio The number of index futures contracts that should be shorted

to hedge an equity portfolio is:

: the parameter beta from the CAPM P: the current value of the portfolio, F=futures price*contract size Daily settlement is taken into considerationAn important assumption of using this formula: the indexhas a beta equal to one.

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F

P N

*

Example


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Example

A fund manager runs a 2,280,000 fund and he is concernedthe market will drop over the next 3 months but believes thatthe market would perform well over the long run. Transactioncosts of selling the portfolio now and buying it back isunacceptably higher than transaction costs in the futuresmarket. The FTSE 100 index is at 5714.3 in the spot market.The 3-month futures price is 5700 and every point in theindex is worth 10. Dividend payment is zero. The fund has a

beta of 1.2.

What strategy would you suggest she to use for hedging theshort term risk of market going down?


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The formula to calculate the required number of contracts whendaily settlement is taken into consideration is:

So 48 index futures needs to be sold in order to hedge the portfolio.

Note: when the market performs poorly, the loss in the portfolio will be compensated by the gain from the futures position.

Is there any assumption that you are making when suggestingto do so?

4810*5700

000,280,2*2.1* F P N


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Reasons for hedging an equity portfolio

The hedger may be1. confident with his well constructed portfolio but

pessimistic about the market

2. planning to hold a portfolio for a long period of timeand requires short-term protection in an uncertainmarket situation.

Note: selling and buying the portfolio back later mightinvolve unacceptably high transaction costs.

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Convenience yield Definition: the benefits from ownership of an asset that are

not obtained by the holder of a long futures contract on theasset.

It reflects the markets expectation concerning the future

availability of the commodity.

The greater the possibility that shortage will occur, thehigher the convenience yield, at which the futures price will

be discounted. Vice versa.


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Mini Quiz

1. The basis strengthens unexpectedly. Which of the followingis true (circle one)

(a) A short hedger's position improves.(b) A short hedger's position worsens.(c) A short hedger's position sometimes worsens and

sometimes improves.(d) A short hedger's position stays the same.

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2. On March 1 the spot price of a commodity is $20 and theJuly futures price is $19. On June 1 the spot price is $24 andthe July futures price is $23.50. A company entered into a

futures contracts on March 1 to hedge the purchase of thecommodity on June 1. It closed out its position on June 1.What is the effective price paid by the company for thecommodity? .

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3. On March 1 the price of a commodity is $300 and the

December futures price is $315. On November 1 the price is

$280 and the December futures price is $281. A producerentered into a December futures contracts on March 1 tohedge the sale of the commodity on November 1. It closedout its position on November 1. What is the effective pricereceived by the producer?

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4. A derivative security:a. is useful only for speculation

b. is useful only for hedgingc. is useful only for arbitraged. can be used for all of these purposese. is useful for none of these purposes

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5. Suppose that the standard deviation of monthly changes in the price of commodity A is $2. The standard deviation of monthlychanges in a futures price for a contract on commodity B (which

is similar to commodity A) is $3. The correlation between thefutures price and the commodity price is 0.9. What hedge ratioshould be used when hedging a one month exposure to the priceof commodity A? ..

6. A company has a $36 million portfolio with a beta of 1.2. The

S&P index is currently standing at 900. Futures contracts on$250 times the index can be traded. What trade is necessary toeliminate all systematic risk in the portfolio. (Indicate thenumber of contracts that should be traded and whether the

position is long or short.)

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Lecture 3 Moodle