1

Futures

Global Financial Management

Campbell R. HarveyFuqua School of Business

Duke Universitycharvey@mail.duke.edu

http://www.duke.edu/~charvey

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Overview

Forward contracts Futures contracts The relationship between forwards and futures Valuation Using forwards and futures to hedge in Practice

» Foreign exchange risk » Stock market risk » Interest rate risk

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Applications for Hedge Instruments

A mining company expects to produce 1000 ounces of gold 2 years from now if it invests in a new mine:» Avoid that the loan for financing the investment cannot be

repaid because the gold price moved A bank expects repayment of a loan in 1 year, and wishes to

use proceeds to redeem 2-year bond» Lock in current interest rate between 1 and 2 years from

now in order to avoid shortfall if interest rates have changed A hotel chain buys hotels in Switzerland, financed with a loan in

US-dollars:» Make sure that the company can repay the loan, even if

Swiss franc proceeds diminished because of exchange rate movement

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Forward Contracts

A forward contract is a contract made today for future delivery of an asset at a prespecified price.» no money or assets change hands prior to maturity.» Forwards are traded in the over-the-counter market.

The buyer (long position) of a forward contract is obligated to:» take delivery of the asset at the maturity date.» pay the agreed-upon price at the maturity date.

The seller (short position) of a forward contract is obligated to:» deliver the asset at the maturity date.» accept the agreed-upon price at the maturity date

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Foreign Exchange Risk:Foreign Currency Futures

Foreign currency futures are traded on the CME.

Foreign currency futures are traded on:» British Pound: 62,500BP » Canadian Dollar: 100,000CD» German Mark: 125,000DM» Japanese Yen: 12,500,000Y» Swiss Franc: 125,000SF» French Franc: 250,000FF» Australian Dollar: 100,000AD» Mexican Peso: 500,000MP

Delivery Months: March, June, Sept., Dec.

Prices are quoted as USD per unit of the foreign currency.» USD/SF = 0.7106» USD/Y = 0.8543» USD/BP = 1.6592» USD/DM = 0.6166

Mar contracts, Open on Tuesday January 21 1997

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Hedging with Foreign Currency Futures

Currency mismatching:» Assets and liabilities in different currencies» Expect receipts and payments in different currencies.» Use currency futures to hedge

Example:

Company has just signed a contract to sell 25 large earth movers to a German mining company:» Total price DM35 million. » Delivery and payment of the earth movers at the end of June.» Current (Jan) USD/DM exchange rate: 0.6136» June futures price for DM: 0.6223» Total outlay today is $21 million; can borrow this amount at 8.4% p. a.

– Is this a worthwhile project?– When would the project make a loss?

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Does the project make a loss?» Borrow $21 million» Repay 1.035*$21m=$21.735m» Make zero profit if USD/DM=0.621

– Do you expect to make a loss or a profit? If you want to hedge:

» receive DM in June» sell a June DM futures contract

– How many contracts do you have to sell? (DM contracts are DM 125,000)

» deliver DM in June at an exchange rate of USD/DM = 0.6223 This will lock in your USD price at:

USD Price = 0.6223 USD/DM (DM35 million)

= $21.7805 million

Hedging with Foreign Currency Futures

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Scenario I: Dollar rises USD/DM exchange rate falls to

0.60 in June The USD price of the earth

movers on the delivery date is:

(DM 35 million)(0.60) = $21m The loss on the project is:

$21m-$21.735m=-$735,000 The profit on the 280 DM futures

contracts is:

DM35m(0.6223-0.60)USD/DM

= $780,500 Total profits are:

$780,500-$735,000=$45,500

Hedging with Foreign Currency Futures

Scenario II: Dollar falls USD/DM exchange rate increases to

0.65 in June The USD price of the earth movers

on the delivery date is:

(DM 35 m)(0.65) = $22.75 m The profit on the project is:

$22.75m-$21.735m=$1.015m The loss on the futures contracts for

DM35m is:

DM35m(0.6223-0.65)USD/DM

=$969,500 Total profits are:

$1,015,000 - $969,500=$45,500

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How to make your own forward contract

Suppose a currency futures contract for the maturity and currency you need does not exist.» Make your own!

The following three transactions replicate the previous contract:» Borrow Deutsche Mark in January to repay in June» Exchange DM proceeds at spot rate into USD» Invest dollars at current interest rate until June

– Why does this work?

January June

USD

DM

Invest at 8.4%

Borrow at 4.93 %

0.6136 0.6223

DM35m

$21.7805m$21.044m

DM34.296m

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Alternatives to Forwards?Homemade forward contracts

Combine borrowing and lending with currency spot contract to replicate forward contract

Other financial policies» The company wanted to borrow dollars anyway? How could

you consolidate the transactions?» What would be the easiest way to avoid the necessity to

hedge?

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Forwards on SecuritiesThe case of a security without income

What is the difference between buying a security today, and between buying a security forward?» If you purchase the security forward, you do not have to pay

the purchase price today:– Can invest the money somewhere else

» Securities pay income (dividends, coupons)– Only if you purchase it now, not if you buy forward

Example:» Share trades today at $25» Pays no dividends during the next three months» The risk free rate for 3 months is 6% p. a. with quarterly

compounding

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What is the forward price? Consider the following two strategies:

» Buy one share for $25» Sell the share forward in three months for the forward price F

What are the cash flows:» Today: Zero from forward, -$25 from buying share» 3 Months: F from selling share forward - value of share

+ value of share = F– Riskless investment of $25 dollars yields F

Investing $25 at riskless rate gives:

$25*1.015=$25.375 Identical portfolios must have same return:

F=$25.375

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Now suppose the share pays a dividend at the end of three months of $2

What are the cash flows now:» Today: Zero from forward, -$25 from buying share» 3 Months: F from selling share forward - value of share

+ value of share +$2 dividend = F+$2– Riskless investment of $25 dollars yields F+$2

Investing $25 at riskless rate gives:

$25*1.015=$25.375 Identical portfolios must have same return:

F+$2=$25.375, hence F=$23.375

Forwards on SecuritiesThe case of a security with income

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The General Formula

Portfolio I:» Buy stock at S0, Sell share forward at F

Portfolio II:» Invest S0 at risk free rate

Cash flows from this are:

Hence we obtain: F= S0(1+rT)-DT

Today 3 Months

Portfolio I Stock -S0 ST+DT

Forward 0 F- ST

Net -S0 F+DT

Portfolio II -S0 S0(1+rT)

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Forward Price and Arbitrage Case 1: F< S0(1+rT)-DT

Then portfolio I has a lower payoff than portfolio II:» Buy portfolio II, (short) sell portfolio I

– Invest in bonds– (short) sell stock– buy stock forward at F

Realize arbitrage profit -F+ S0(1+rT)-DT>0

Today 3 MonthsSell Stock S0 -(ST+DT)Buy Forward 0 ST-FSell Bond -S0 +S0(1+r)Total 0 -F-DT

S0(1+r)

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The Standard Formula

Previous formula unusual.» Assume you receive dividend up front:

» Rewrite dividend as dividend yield d:

» Then the previous formula can be rewritten:

» The conventional way to express this is (use continuous compounding):

D D rT T01

1

D d S0 0 *

F S r D

S d rT T

T

0

0

1

1 1

( )

( )

F S e r d T 0

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Futures Contracts

A futures contract is identical to a forward contract, except for the following differences:» Futures contracts are standardized contracts and are traded on

organized exchanges.» Futures contracts are marked-to-market daily.» The daily cash flows between buyer and seller are equal to the

change in the futures price. Futures and forward prices must be identical if interest rates are

constant.» Can use results on forward for futures

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Futures Contracts

Futures contracts allow investors to: » Hedge» Speculate

Futures contracts are available on commodities and financial assets:» Agricultural products and livestock» Metals and petroleum» Interest rates» Currencies» Stock market indices

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Valuation of Futures ContractsAn Application

A futures contract on the S&P500 Index entitles the buyer to receive the cash value of the S&P 500 Index at the maturity date of the contract.

The buyer of the futures contract does not receive the dividends paid on the S&P500 Index during the contract life.

The price paid at the maturity date of the contract is determined at the time the contract is entered into. This is called the futures price.

There are always four delivery months in effect at any one time.» March» June » September» December

The closing cash value of the S&P500 Index is based on the opening prices on the third Friday of each delivery month.

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Hedging Stock Market Risk: S&P500 Futures Contract

Contract: S&P500 Index Futures Exchange: Chicago Merchantile Exchange Quantity: $500 times the S&P 500 Index Delivery Months: March, June, Sept., Dec. Delivery Specs: Cash Settlement Based on the

Value of the S&P 500 Index at

Maturity. Min. Price Move: 0.05 Index Pts. ($25 per

contract).

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Valuation of Futures ContractsAn Alternative Derivation

When you buy a futures contract on the S&P500 Index, your payoff at the maturity date, T, is the difference between the cash value of the index, ST, and the futures price, F.

The amount you put up today to buy the futures contract is zero. This means that the present value of the futures contract must also be zero:

The present value of ST and F is:

Then, using the fact that PV(F)=PV(ST):

Payoff S FT

PV S F PV S PV FT T( ) ( ) ( ) 0

PV S S PV Div S e

PV F Fe

TdT

rT

( ) ( )

( )

0 0

F S e r d T 0

( )

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Example

On Thursday January 22, 1997 we observed:» The closing price for the S&P500 Index was 786.23. » The yield on a T-bill maturing in 26 weeks was 5.11%» Assume the annual dividend yield on the S&P500 Index is

1.1% per year, – What is the futures price for the futures contracts maturing

in March, June, September, December 1997?

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Example

Days to maturity» June contract: 148 days

Estimated futures prices:» For the June contact:

» Similarly:

F S e

e

Juner d T

0

0 0511 0 011 148 365786 23 799 12. .( . . )( / )

Maturity Days Price Actual PriceMarch 57 791.17 791.6June 148 799.12 799.0September 239 807.15 806.8December 330 815.26 814.8

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Index Arbitrage Suppose you observe a price of 820 for the June 1997 futures contract.

How could you profit from this price discrepancy? We want to avoid all risk in the process.

Buy low and sell high:

» Borrow enough money to buy the index today and immediately sell a June futures contract at a price of 820.

» At maturity, settle up on the futures contract and repay your loan.Position 0 TBorrow 782.73 -799.12Buy e(-dT) units of index -782.73 STSell 1 futures contract 0.00 820-STNet position 0.00 20.88

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Index Arbitrage Suppose the futures price for the September contract was 790. How

could you profit from this price discrepancy? Buy Low and Sell High:

» Sell the index short and use the proceeds to invest in a T-bill. At the same time, buy a September futures contract at a price of 790.

» At settlement, cover your short position and settle your futures position.

Position 0 TLend -780.59 807.15Sell e(-dT) units of index 780.59 -STBuy 1 futures contract 0 ST-790Net position 0.00 17.15

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Hedging with S&P500 Futures Suppose a portfolio manager holds a portfolio that mimics the S&P500

Index.» Current worth: $99.845 million, up 20% through mid-November ‘95 » S&P500 Index currently at 644.00» December S&P500 futures price is 645.00.

– How can the fund manager hedge against further market movements?

Lock in a price of 645.00 for the S&P500 Index by selling S&P500 futures contracts.» Lock in a total value for the portfolio of:

$99.845(645.00/644.00) million = $100.00 million. Since one futures contract is worth $500(645.00) = $322,500, the total

number of contracts that need to be sold is:

100 00

322 500310 08

.

,.

million

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Hedging with S&P500 Futures

Scenario I: Stock market falls

Suppose the S&P500 Index falls to 635.00 at the maturity date of the futures contract.

The value of the stock portfolio is:

99.845(635.00/644.00) = 98.45 million

The profit on the 310 futures contracts is:

310(500)(645.00-635) = 1.55 million

The total value of the portfolio at maturity is $100 million.

Scenario II: Stock market rises

Suppose the S&P500 Index increases to 655.00 at the maturity date of the futures contract.

The value of the stock portfolio is:

99.845(655.00/644.00) = 101.55 million

The loss on the 310 futures contracts is:

310(500)(645.00-655.00) = -1.55 million

The total value of the portfolio at maturity is $100 million.

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Commodity Futures Commodities are similar in many ways to securities, but some

important differences:» Storage costs can be significant:

– Security (precious metals)– Physical storage (grain)– Possibility of damage

Summarized as cost of carry, usually written as constant annual percentage q of initial value.

» Sometimes possession of commodity also provides benefits:– Demand fluctuations– Supply shortages (Oil)

Summarized as convenience yield, usually written as constant annual percentage y of initial value.

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Example: Cost of Carry

You are considering taking physical delivery of live cattle in order to execute a commodity futures arbitrage.

The cost of carry is assessed at 4% relative to the current spot price of $100.

If the contract has 2 months to maturity, the up-front cost of storing and feeding the cattle is:

CC = S0(eqT-1) = 100(e0.04(2/12)-1) =$0.669.

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Replication of Forward Contracts

The payoff of a forward contract can be replicated by » borrowing money» buying the commodity» paying the cost of carry (feed for hogs, security for gold,

storage for oil) Two Implications:

» If two procedures generate the same cash flows, they must cost the same

» If an appropriate forward contract does not exist, we can make our own by:– transacting in the spot market and– borrowing

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Valuation of Commodity Contracts

Position Initial Cash Flow Terminal Cash FlowBuy one unit of commodityPay cost of carryBorrowEnter forward saleNet portfolio value

-S0 ST

F-ST

0

0

-S0(eqT-1)S0eqT

0

-S0e(q+r)T

F-S0e(q+r)T

In the absence of arbitrage: F = S0e(q+r)T

Compare this with formula for dividend paying stocks:» cost of carry is like negative dividend» same principles for valuation apply

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Example: Forward Arbitrage

The spot price of wheat is 550 and the six-month forward price is 600. The riskless rate of interest is 5% p.a. and the cost of carry is 6% p.a.

Is there an arbitrage opportunity in this market?

Position Initial Cash Flow Terminal Cash FlowBuy one unit of commodityPay cost of carryBorrowEnter forward saleNet portfolio value

-550 ST

600-ST

0

0

-550(e0.06(0.5)-1)550e0.06(0.5)

0

-550e(0.06+0.05)0.5

600-550e(0.06+0.05)0.5

Arbitrage Profit: 600-550e(0.06+0.05)0.5 = $18.90

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Hedging Using Interest Rate Futures Contracts

Hedging interest rate risk can also be done by using interest rate futures contracts.

There are two main interest rate futures contracts:» Eurodollar futures» US T-bond futures

The Eurodollar futures is the most popular and active contract. Open interest is in excess of $4 trillion at any point in time.

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LIBOR The Eurodollar futures contract is based on the interest rate payable on a

Eurodollar time deposit. This rate is known as LIBOR (London Interbank Offer Rate) and has

become the benchmark short-term interest rate for many US borrowers and lenders.

Eurodollar time deposits are non-negotiable, fixed rate US dollar deposits in offshore banks (i.e., those not subject to US banking regulations).

US banks commonly charge LIBOR plus a certain number of basis points on their floating rate loans.

LIBOR is an annualized rate based on a 360-day year. Example: The 3-month (90-day) LIBOR 8% interest on $1 million is

calculated as follows:.($1, , ) $20,

08

4000 000 000

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Eurodollar Futures Contract

The Eurodollar futures contract is the most widely traded short-term interest rate futures.

It is based upon a 3-month $1 million Eurodollar time deposit. It is settled in cash. At expiration, the futures price is 100-LIBOR. Prior to expiration, the quoted futures price implies a LIBOR rate of:

Implied LIBOR = 100-Quoted Futures Price

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Eurodollar Futures Contract

Contract: Eurodollar Time Deposit Exchange: Chicago Merchantile Exchange Quantity: $1 Million Delivery Months: March, June, Sept., and Dec. Delivery Specs: Cash Settlement Based on

3-Month LIBOR Min Price Move: $25 Per Contract (1 Basis Pt.)

( / )( , , )

$251 100 1%)($1 000 000

4

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Example

Suppose in February you buy a March Eurodollar futures contract. The quoted futures price at the time you enter into the contract is 94.86.

If the LIBOR rate falls 100 basis points between February and the expiration date of the contract in March, what is your profit or loss?

The quoted price at the time the contract is purchased implies a LIBOR rate of 100-94.86 = 5.14%.

If LIBOR falls 100 basis points, it will be 4.14% at the expiration date of the contract.

This means a futures price of 100-4.14 = 95.86 at the expiration date. Since we bought the contract at a futures price of 94.86, our total gain is

95.86-94.86 = 1.00.

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Example

In dollar terms, our gain is:

The increase in the futures price is multiplied by $10,000 because the futures price is per $100 and the contract is for $1,000,000.

We divide the increase in the futures price by 4 because the contract is a 90 day (3 month) contract.

Gain ( . . )( , )$2,

9586 94 86 10 000

4500

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Hedging with Eurodollar Futures Contracts (1)

Suppose a firm knows in February that it will be required to borrow $1 million in March for a period of 3 months (90 days).» The rate that the firm will pay for its borrowing is LIBOR + 50 basis

points.» The firm is concerned that interest rates may rise before March

and would like to hedge this risk.» Assume that the March Eurodollar futures price is 94.86.» The LIBOR rate implied by the current futures price is 100-94.86 =

5.14%. If the LIBOR rate increases, the futures price will fall. Therefore, to

hedge the interest rate risk, the firm should sell one March Eurodollar futures contract.

The gain (loss) on the futures contract should exactly offset any increase (decrease) in the firm’s interest expense.

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Hedging with Eurodollar Futures Contracts (3)

Suppose LIBOR increases to 6.14% at the maturity date of the futures contract.

The interest expense on the firm’s $1 million loan commencing in March will be:

The gain on the Eurodollar futures contract is:

(. . )($1, , )$16,

0614 005 000 000

4600

( . . )( , )$2,

94 86 9386 10 000

4500

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Hedging with Eurodollar Futures Contracts (4)

Now assume that the LIBOR rate falls to 4.14% at the maturity date of the contract.

The interest expense on the firm’s $1 million loan commencing in March will be:

The gain on the Eurodollar futures contract is:

(. . )($1, , )$11,

0414 005 000 000

4600

( . . )( , )$2,

94 86 9586 10 000

4500

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Hedging with Eurodollar Futures Contracts (5)

The net outlay is equal to $14,100 regardless of what happens to LIBOR.

This is equivalent to paying 5.64% (1.41% for 3 months) on $1 million. The 5.64% borrowing rate is equal to the current LIBOR rate of 5.14%,

plus the additional 50 basis points that the firm pays on its short-term borrowing.

The firm’s futures position has locked in the current LIBOR rate.

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Alternatives to Interest Rate Forwards

Is there a homemade forward?

We saw that we can generally replicate futures and forward contracts by:» Buying and selling in the spot market» Borrowing

Suppose an appropriate forward/futures contract does not exist» Can we make our own forward» Which instruments should we use? How?

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Using Bonds to Hedge Interest Risk

An Example of homemade interest rate forwards

Suppose you expect:» Receipt of $1 million exactly one year from today » Need it to repay a loan exactly two years from today.» You would like to invest the $1 million between years 1 and 2.

Issues:» Why is this risky?» How can you lock in the interest rate?» Which interest rate do you want to lock in?» What can you do if a suitable futures contract with the appropriate

currency and maturity does not exist?* Use implied forward rates

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Use Implied Forward Rates to Hedge

To lock-in the interest rate on your $1 million you need to buy a two-year zero-coupon bond and sell a one-year zero-coupon bond.

The exact transaction involves selling $1/(1+r1) million of the one-year zero-coupon bond and using the proceeds to purchase a two-year zero-coupon bond yielding r2.

This transaction will lock-in an interest rate of 1f1 over the second year on your $1 million.

Example:» r1 =4.81%

» r2 =4.94%

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Using Implied Forward Rates to Hedge

The one-year forward rate is 1f1 = 5.07%.

Position 0 1 2

Sell 1-year zero(r1 = 4.81%)

$954,107 -1,000,000 -

Buy 2-year zero(r2 = 4.94%)

-$954,107 - 1,050,702

Cash Receipt - 1,000,000 -

Net Position 0 0 1,050,702

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Using Implied Forward Rates to Hedge

Another Example

Now suppose you expect to receive $1 million two years from today and need the money to pay a debt exactly five years from today. How can you lock in the interest rate on your $1 million?

The exact transaction involves selling $1/(1+r2)2 million of the two-year zero-coupon bond and using the proceeds to buy a five-year zero-coupon bond yielding r5.

This transaction locks in an interest rate of 2f3 on your $1 million.

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Using Implied Forward Rates to Hedge

Example II

The 3-year forward rate starting 2 years from now is denoted 2f3 and is computed as follows:

[The notation here is different from lecture note where this would be 2f5]

Using the spot rates in effect on 2/6/96, we have:

2 35

5

22

1 31

11f

r

r

( )

( )

/

2 3

5

2

1 310527

104941 5 49%f

( . )

( . ).

/

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The final value of the investment should be $1(1+2f3)3 = $1(1.0549)3 = $1,173,927.

Position 0 2 5

Sell 2-year zero(r2= 4.94%)

$908,067 -1,000,000 -

Buy 5-year zero(r5 = 5.27%)

-$908,067 - 1,173,927

Cash Receipt - 1,000,000 -

Net Position 0 0 1,173,927

Using Implied Forward Rates to Hedge

Example II

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Relationship Between Forward Rates:

The General Case The t-year forward rate starting n years from today is equivalent to earning

the one-year forward rates between year n and year n+t.

n n+1 n+2 ... n+t-1 n+t

nf1 n+1f1 ... n+t-1f1

nft

Suppose you wanted to know the implicit interest rate you would earn between year n and year n+t. This involves calculating the t-year forward rate starting n years from today.

The general formula is:

n t ii n

n t t

f f

( )/

1 11

1 1

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Relationship Between Forward Rates:

Example II

Use the spot rates on 2/6/96 to calculate the one-year forward rates in years 3-5. How does this compare to 2f3?

Maturity(Years)

Spot Rate(rt)

1-YearForward Rate

(tf1)2 4.94% -

3 5.06% 5.30%

4 5.18% 5.54%

5 5.27% 5.63%

2 31 31053 10554 10563 1 5 49%f [( . )( . )( . )] ./

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Summary

Forward and futures can be used to hedge risks:» Foreign exchange rate risk» Stock market risk» Commodity price risk» Interest rate risk

Forwards and futures are redundant instruments:» Can be replicated through transactions in spot markets and

borrowing or lending Forwards and futures are derivatives:

» Value depends on value of other asset