Class 4

Forward and Futures Contracts

Overview

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

Interest rate risk Foreign exchange risk Stock market risk

Forward Contracts

A forward contract is a contract made today for future delivery of an asset at a prespecified price.

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.

Forward Contracts

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.

With a forward contract, no money or assets change hands prior to maturity.

Forwards are traded in the over-the-counter market.

Using Forward Rates to Hedge

Suppose you expect to recieve $1 million exactly one year from today and need it to pay a debt exactly two years from today. You would like to invest the $1 million between years 1 and 2, but are concerned that interest rates may fall between now and then. How can you lock in the interest rate on your $1 million?

Using 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.

Using 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

Forward Rates: General Formula

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 formula is:

n tt n

n t

nn

t

fr

r

( )

( )

/1

11

1

Using Forward Rates to Hedge 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.

Using Forward Rates to Hedge

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

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

( . )

( . ).

/

Using Forward Rates to Hedge 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

Relationship Between Forward Rates

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

Relationship Between Forward Rates

Mathematically, this relationship can be expressed as:

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?

n t i

i n

n t t

f f

( )

/

1 11

1 1

Relationship BetweenForward Rates

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 [( . )( . )( . )] ./

Valuation 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) If two procedures generate the same cash

flows, they must cost the same

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.

Valuation of Forward 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

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?

Example: Forward Arbitrage

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

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.

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 indicies

Valuation of Futures Contracts

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.

Valuation of Futures Contracts

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.

Valuation of Futures Contracts

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.

Payoff S FT

Valuation of Futures Contracts The futures price, F, must satisfy:

The present value of ST is:

The present value of F is:

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

PV S S PV Div S eTdT( ) ( )

0 0

PV F Fe rT( )

Valuation of Futures Contracts This yields the following relationship between

the futures price and the cash price:

F S e S er d T bT 0 0

( )

Example

The closing price for the S&P500 Index on Wednesday January 31, 1996 was 636.00. The yield on a T-bill maturing in 365 days was 4.88% on that day. If the annual dividend yield on the S&P500 Index is 2.0% per year, what is the futures price for the contract maturing in June 1996? What about a futures contract maturing in December 1996?

Example Days to maturity

June contract: 141 days December contract: 323 days

Estimated futures prices:

F eJune 636 643120 0488 0 02 141 365( . . )( / ) .

F eDec 636 652 420 0488 0 02 323 365( . . )( / ) .

Index Arbitrage

Suppose you observe a price of 650 for the June 1996 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 650.

At maturity, settle up on the futures contract and repay your loan.

Index Arbitrage

The cash flows from this transaction are outlined below.

Position 0 T

Borrow 631.11 642.92

Buy e-dT

Units of Index -631.11 ST

Sell 1 Futures Contract 0 650-ST

Net Position 0 7.08

Index Arbitrage

Note that the transaction involves a zero cash outlay today.

The transaction involves a sure cash flow at date T of:

Cash Flow at T = F-S0e(r-d)T

= 650 - 636e(0.048-0.02)(141/365)

= 7.08 This is an arbitrage profit: A positive profit

with no risk and no investment.

Index Arbitrage

Suppose the futures price for the December contract was 645. 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 December futures contract at a price of 645.

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

Index Arbitrage

The cash flows from this transaction are outlined below.

Position Initial Cash Flow Terminal Cash FlowLend -624.84 651.96Sell exp(-dT) units of index 624.84Buy one futures contract 0 -645Net Position 0 6.96ST

-ST

Index Arbitrage

Note that the transaction involves a zero cash outlay today.

The transaction involves a sure cash flow at date T of:

Cash Flow at T = S0- Fe(r-d)T

= 636e(0.048-0.02)(323/365)-645

= 6.96 This again represents an arbitrage profit.

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.

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).

LIBOR

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

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

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.)

( / )( , , )$25

1 100 1%)($1000 000

4

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?

Example

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.

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

( . . )( , )

$2,95 86 94 86 10 000

4500

Hedging with Eurodollar Futures Contracts

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.

Hedging with Eurodollar Futures Contracts

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.

Hedging with Eurodollar Futures Contracts

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 93 86 10 000

4500

Hedging with Eurodollar Futures Contracts

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 95 86 10 000

4500

Hedging with Eurodollar Futures Contracts

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.

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).

S&P500 Futures Contract

On February 7, 1996, the S&P500 Index closed at 649.93. On the same day the June S&P500 Index price was 662.00. If you buy the June S&P500 futures contract, what is your gain or loss if the S&P500 Index closes at 658.50 on the expiration date of the contract?

Pr $500( . . ) $1,ofit 658 50 662 00 750

Hedging with S&P500 Futures

Suppose a portfolio manager holds a portfolio that mimics the S&P500 Index. The fund is currently worth $99.845 million and is up 20% through mid-November. The S&P500 Index currently stands at 644.00 and the December S&P500 futures price is 645.00. If the fund manager wishes to hedge against further market movements, how can this be done?

Hedging with S&P500 Futures The fund manager can lock in a price of 645.00

for the S&P500 Index by selling S&P500 futures contracts. This will 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

Hedging with S&P500 Futures

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.

Hedging with S&P500 Futures

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 profit 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.

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

Foreign Currency Futures

Delivery Months: March, June, Sept., Dec. Prices are quoted as USD per unit of the foreign

currency. USD/SF = 0.8335 USD/Y = .009493 USD/BP = 1.5466 USD/DM = 0.6821

Hedging with Foreign Currency Futures

Suppose your company has just signed a contract to sell a German mining company 25 large earth movers for a total price of DM35 million. Delivery and payment of the earth movers will take place at the end of June. The current USD/DM exchange rate is 0.6789 and the June futures price for DM is 0.6821. If you are worried about exchange rate movements between now and June, how can you hedge your risk and lock in the USD price of the earth movers?

Hedging with Foreign Currency Futures

Since you will receive DM in June, you need to sell a June DM futures contract. This requires you to deliver DM in June at an exchange rate of USD/DM = 0.6821.

This will lock in your USD price at:

USD Price = 0.6821(DM35 million)

= $23.8735 million The number of contracts you need to sell is

equal to (DM35 million)/125,000 = 280 contracts.

Hedging With ForeignCurrency Futures

Suppose the USD/DM exchange rate falls to 0.6530 at the maturity of the futures contract.

The USD price of the earth movers on the delivery date is:

(DM 35 million)(0.6530) = $22.855 million The profit on the 280 DM futures contracts is:

(280)(125,000)(0.6821-0.6530) = $1.0185 million The total USD cash flow is $23.8735 million.

Hedging With ForeignCurrency Futures

Suppose the USD/DM exchange rate increases to 0.6950 at the maturity of the futures contract.

The USD price of the earth movers on the delivery date is:

(DM 35 million)(0.6950) = $24.325 million The profit on the 280 DM futures contracts is:

(280)(125,000)(0.6821-0.6950) = -$0.4515 million The total USD cash flow is $23.8735 million.