Futures
Global Financial Management
Campbell R. Harvey
Fuqua School of Business
Duke University
charvey@mail.duke.edu
http://www.duke.edu/~charvey
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Overview
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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
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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
3
Forward Contracts
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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
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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
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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
5
Hedging with Foreign Currency Futures
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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?
6
Hedging with Foreign Currency Futures
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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
7
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
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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
8
How to make your own forward
contract
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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
USD
Invest at 8.4%
$21.044m
0.6136
DM
June
$21.7805m
0.6223
DM34.296m
Borrow at 4.93 %
DM35m
9
Alternatives to Forwards?
Homemade forward contracts
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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 Securities
The case of a security without income
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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?
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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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Forwards on Securities
The case of a security with income
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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
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The General Formula
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Portfolio I:
» Buy stock at S0, Sell share forward at F
Portfolio II:
» Invest S0 at risk free rate
Cash flows from this are:
Portfolio I
Portfolio II
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Today
3 Months
Stock
-S0
ST+DT
Forward
0
F- ST
Net
-S0
F+DT
-S0
S0(1+rT)
Hence we obtain: F= S0(1+rT)-DT
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Forward Price and Arbitrage
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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
Sell Stock
Buy Forward
Sell Bond
Total
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Today
S0
0
-S0
0
3 Months
-(ST+DT)
ST-F
+S0(1+r)
-F-DT
S0(1+r)
Realize arbitrage profit -F+ S0(1+rT)-DT>0
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The Standard Formula
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Previous formula unusual.
» Assume you receive dividend up front:
D0  DT 1  rT 
1
» Rewrite dividend as dividend yield d:
D0  d * S0
» Then the previous formula can be rewritten:
F  S0 (1  rT )  DT
 S0 1  d (1  rT )
» The conventional way to express this is (use continuous
compounding):
F  S0e r d T
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Futures Contracts
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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
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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 Contracts
An Application
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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
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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 Contracts
An Alternative Derivation
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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.
Payoff  ST  F
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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:
PV ( ST  F )  0  PV ( ST )  PV ( F )
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The present value of ST and F is:
PV ( ST )  S0  PV ( Div )  S0e dT
PV ( F )  Fe rT
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Then, using the fact that PV(F)=PV(ST):
F  S0 e ( r  d ) T
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Example
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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
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Days to maturity
» June contract: 148 days
Estimated futures prices:
» For the June contact:
FJune  S0e r d T
 786.23e ( 0.05110.011)(148/ 365)  799.12
» Similarly:
Maturity
March
June
September
December
Days
57
148
239
330
Price
Actual Price
791.17
791.6
799.12
799.0
807.15
806.8
815.26
814.8
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Index Arbitrage
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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.
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Buy low and sell high:
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» Borrow enough money to buy the index today and immediately sell a
June
futures contract at a price0of 820.
Position
T
» At
maturity, settle up on the futures
contract-799.12
and repay your loan.
Borrow
782.73
Buy e(-dT) units of index
Sell 1 futures contract
Net position
-782.73
0.00
0.00
ST
820-ST
20.88
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Index Arbitrage
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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
Lend
Sell e(-dT) units of index
Buy 1 futures contract
Net position
0
-780.59
780.59
0
0.00
T
807.15
-ST
ST-790
17.15
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Hedging with S&P500 Futures
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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:
100.00million
 310.08
322,500
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Hedging with S&P500 Futures
Scenario I: Stock market falls
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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
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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
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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
l 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)
l Summarized as convenience yield, usually written as
constant annual percentage y of initial value.
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Example: Cost of Carry
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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
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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
Buy one unit of commodity
Pay cost of carry
Borrow
Enter forward sale
Net portfolio value
Initial Cash Flow
-S0
-S0(eqT-1)
S0eqT
0
0
Terminal Cash Flow
ST
0
-S0e(q+r)T
F-ST
F-S0e(q+r)T
In the absence of arbitrage: F = S0e(q+r)T
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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
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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
Buy one unit of commodity
Pay cost of carry
Borrow
Enter forward sale
Net portfolio value
Initial Cash Flow
Terminal Cash Flow
-550
ST
-550(e0.06(0.5)-1)
0
-550e(0.06+0.05)0.5
550e0.06(0.5)
0
600-ST
0
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
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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
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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:
.08
($1,000,000)  $20,000
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Eurodollar Futures Contract
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The Eurodollar futures contract is the most widely traded short-term
interest rate futures.
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It is based upon a 3-month $1 million Eurodollar time deposit.
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It is settled in cash.
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At expiration, the futures price is 100-LIBOR.
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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
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Contract: Eurodollar Time Deposit
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Exchange: Chicago Merchantile Exchange
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Quantity: $1 Million
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Delivery Months: March, June, Sept., and Dec.
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Delivery Specs: Cash Settlement Based on
3-Month LIBOR
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Min Price Move: $25 Per Contract (1 Basis Pt.)
(1 / 100)(1%)($1,000,000)
 $25
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Example
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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.
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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?
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The quoted price at the time the contract is purchased implies a LIBOR
rate of 100-94.86 = 5.14%.
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If LIBOR falls 100 basis points, it will be 4.14% at the expiration date of
the contract.
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This means a futures price of 100-4.14 = 95.86 at the expiration date.
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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
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In dollar terms, our gain is:
Gain 
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(95.86  94.86)(10,000)
 $2,500
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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.
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Hedging with Eurodollar
Futures Contracts (1)
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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)
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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:
(.0614 .005)($1,000,000)
 $16,600
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The gain on the Eurodollar futures contract is:
(94.86  9386
. )(10,000)
 $2,500
4
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Hedging with Eurodollar
Futures Contracts (4)
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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:
(.0414 .005)($1,000,000)
 $11,600
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The gain on the Eurodollar futures contract is:
( 94.86  95.86)(10,000)
 $2,500
4
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Hedging with Eurodollar
Futures Contracts (5)
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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?
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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
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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
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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 zerocoupon 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
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The one-year forward rate is 1f1 = 5.07%.
Position
0
1
2
Sell 1-year zero
(r1 = 4.81%)
Buy 2-year zero
(r2 = 4.94%)
Cash Receipt
$954,107
-1,000,000
-
-$954,107
-
1,050,702
-
1,000,000
-
Net Position
0
0
1,050,702
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Using Implied Forward Rates to Hedge
Another Example
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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?
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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.
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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
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The 3-year forward rate starting 2 years from now is denoted 2f3 and
is computed as follows:
 (1  r5 ) 
f3  
2
 (1  r2 ) 
5
2
1/ 3
1
[The notation here is different from lecture note where this would be 2f5]
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Using the spot rates in effect on 2/6/96, we have:
2
 (10527
.
)5 
f3  

.
)2 
 (10494
1/ 3
 1  5.49%
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Using Implied Forward Rates to Hedge
Example II
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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%)
Buy 5-year zero
(r5 = 5.27%)
Cash Receipt
$908,067
-1,000,000
-
-$908,067
-
1,173,927
-
1,000,000
-
Net Position
0
0
1,173,927
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Relationship Between Forward Rates:
The General Case
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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
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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:
1/ t
n  t 1

1
n f t    (1 i f1 )
 in

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Relationship Between Forward Rates:
Example II
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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)
2
Spot Rate
(rt)
4.94%
1-Year
Forward Rate
( tf 1 )
-
3
5.06%
5.30%
4
5.18%
5.54%
5
5.27%
5.63%
2
f 3  [(1053
.
)(10554
.
)(10563
.
)]1/ 3  1  5.49%
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Summary
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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
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