Chapter 11

Forwards and Futures

FIXED-INCOME SECURITIES

Outline

• Futures and Forwards• Types of Contracts• Trading Mechanics• Trading Strategies• Futures Pricing• Uses of Futures

Futures and Forwards

• Forward– An agreement calling for a future delivery of an asset at an agreed-

upon price

• Futures– Similar to forward but feature formalized and standardized

characteristics

• Key differences in futures– Secondary market - liquidity

– Marked to market

– Standardized contract units

– Clearinghouse warrants performance

Key Terms for Futures Contracts

• Futures price: agreed-upon price (similar to strike price in option markets)

• Positions– Long position - agree to buy

– Short position - agree to sell

• Interpretation– Long : believe price will rise (or want to hedge price decline)

– Short : believe price will fall (or want to hedge price increase)

• Profits on positions at maturity (zero-sum game)– Long = spot price ST minus futures price F0

– Short = futures price F0 minus spot price ST

Markets for Interest Rates Futures

• The International Money Market of the Chicago Mercantile Exchange (www.cme.com)

• The Chicago Board of Trade (www.cbot.com)• The Sydney Futures Exchange• The Toronto Futures Exchange• The Montréal Stock Exchange• The London International Financial Futures Exchange (

www.liffe.com)• The Tokyo International Financial Futures Exchange (TIFFE)• Le Marché à Terme International de France (www.matif.fr)• Eurex (www.eurexchange.com)

InstrumentsCME CBOT LIFFE

Eurodollar Futures 30-Year US Treasury Bonds Long Gilt Contract13-Week Treasury Bill Futures 10-Year US Treasury Notes German Government Bond

ContractLibor Futures 5-Year US Treasury Notes Japanese Government Bond

ContractFed Funds Turn Futures 2-Year US Treasury Notes 3-Month Euribor Future10-Year Agency Futures 10-Year Agency Notes 3-Month Euro Libor Future5-Year Agency Futures 5-Year Agency Notes 3-Month Sterling Future

Argentine 2X FRB Brady BondFutures

Long Term Municipal BondIndex

3-Month Euro Swiss FrancFuture

Argentine Par Bond Futures 30-Day Federal Funds Mortgage 3-Month Euroyen (TIBOR)Future

Brazilian 2 X C Brady BondFutures

3-Month Euroyen (LIBOR)Future

Brazilian 2 X EI Brady BondFutures

2-Year Euro Swapnote

Mexican 2 X Brady BondFutures

5-Year Euro Swapnote

Euro Yen Futures 10-Year Euro SwapnoteJapanese Government Bond

FuturesEuro Yen Libor FuturesMexican TIIE Futures

Mexican CETES Futures

Characteristics of Future Contracts

• A future contract is an agreement between two parties

• The characteristics of this contract are– The underlying asset

– The contract size

– The delivery month

– The futures price

– The initial regular margin

Underlying Asset and Contract Size

• The underlying asset that the seller delivers to the buyer at the end of the contract may exist (interest rate) or may not exist (bond)

– The underlying asset of the CBOT 30-Year US Treasury bond future is a fictive 30-year maturity US Treasury bond with 6% coupon rate

• The contract size specifies the notional principal or principal value of the asset that has to be delivered

– The notional principal of the CBOT 30-Year US Treasury bond future is $100,000

– The principal value of the Matif 3-month Euribor Future to be delivered is euros 1,000,000

Price

• The futures price is quoted differently depending on the nature of the underlying asset– When the underlying asset is an interest rate, the future price is

quoted to the third decimal point as 100 minus this interest rate

– When the underlying asset is a bond, it is quoted in the same way as a bond, i.e., as a percentage of the nominal value of the underlying

• The tick is the minimum price fluctuation that can occur in trading

• Sometimes daily price movement limits as well as position limits are specified by the exchange

Trading Arrangements

• Clearinghouse acts as a party to all buyers and sellers– Obligated to deliver or supply delivery

• Initial margin– Funds deposited to provide capital to absorb losses

• Marking to market– Each day profits or losses from new prices are reflected in the account

• Maintenance or variation margin– An established margin below which a trader’s margin may not fall

• Margin call – When the maintenance margin is reached, broker will ask for additional

margin funds

Conversion Factor

• When the underlying asset of a future contract does not exist, the seller of the contract has to deliver a real asset

– May differ from the fictitious asset in terms of coupon rate – May differ from the fictitious asset in terms of maturity

• Conversion factor tells you how many units of the actual asset are worth as much as one unit of the fictive underlying asset

• Given a future contract and an actual asset to deliver, it is a constant factor which is known in advance

• Conversion factors for next contracts to mature are available on web sites of futures markets

Conversion Factor (Cont’)

• Consider – A future contract whose fictitious underlying asset is a m year maturity bond

with a coupon rate equal to r – Suppose that the actual asset delivered by the seller of the future contract is

a x-year maturity bond with a coupon rate equal to c

• Expressed as a percentage of the nominal value, the conversion factor denoted CF is the present value at maturity date of the future contract of the actual asset discounted at rate r

• Example– Consider a 1 year future contract whose underlying asset is a fictitious 10-

year maturity bond with a 6% annual coupon rate– Suppose that the asset to be delivered is at date 1 a 10-year maturity bond

with a 5% annual coupon rate

3991.926$

%61

000,1

%)61(

5010

110

i

iCF

Invoice Price

• The conversion factor is used to calculate the invoice price – Price the buyer of the future contract must pay the seller when a

bond is delivered

– IP = size of the contrat x [futures price x CF]

• Example– Suppose a future contract whose contract size is $100,000, the

future price is 98. The conversion factor is equal to 106.459 and the accrued interest is 3.5.

– The invoice price is equal to

IP = $100,000 x [ 98% x106.459% + 3.5% ] = $104,329.82

Cheapest-to-Deliver

• At the repartition date, there are in general many bonds that may be delivered by the seller of the future contract

• These bonds vary in terms of maturity and coupon rate• The seller may choose which of the available bonds is the

cheapest to deliver• Seller of the contract delivers a bond with price CP and receives

the invoice price IP from the buyer• Objective of the seller is to find the bond that achieves

Max (IP - CP) = Max (futures price x CF – quoted price)

Cheapest-to-Deliver (Example)

• Suppose a future contract – Contract size = $100,000 – Price= 97

• Three bonds denoted A,B and C

• Search for the bond which maximises the quantity IP-CP• Cheapest to deliver is bond C

Quoted Price Conversion Factor IP-CP

Bond A 103.90 107.145% 3,065$

Bond B 118.90 122.512% -6,336$

Bond C 131.25 135.355% 4,435$

Forward Pricing

• Consider at date t an investor who wants to hold at a future date T one unit of a bond with coupon rate c and time t price Pt

• He faces the following alternative cash flows– Either he buys at date t a forward contract from a seller who will

deliver at date T one unit of this bond at a fixed price Ft

– Or he borrows money at a rate r to buy this bond at date t

Date t T

Buy a forward contract written on 1 unit of bond B 0 Ft

Borrow money to buy 1 unit of bond B Pt -Pt 1 r T t360

Buy 1 unit of bond B Pt AC 100 c T t365

-

Forward Pricing

• Note that both trades have a cost equal to zero at date t.• Note also that the position at the end is the same (one unit of

bond).• Then in the absence of arbitrage opportunities, the value of the

two strategies at date t must be equal• From this we obtain

365

100360

1tT

ctT

rPF tt

365

1tT

CRPF tt

oror

with R = 365r/360 and C = 100c/Pt

Forward Pricing - Example

• On 05/01/01, we consider a forward contract maturing in 6 months, written on a bond whose coupon rate and price are respectively 10% and $115

• Assuming a 7% interest rate, the forward price F05/01/01 is equal to

07.114365

184%10100

360

184%7111501/01/05

F

Forward Pricing – Underlying is a Rate

• Simply determine the forward rate that can be guaranteed now on a transaction occurring in the future

• Example– An investor wants now to guarantee the one-year zero-coupon rate for a

$10,000 loan starting in 1 year• Either he buys a forward contract with $10,000 principal value maturing

in 1 year written on the one-year zero-coupon rate R(0,1) at a determined rate F(0,1,1), which is the forward rate calculated at date t=0, beginning in 1 year and maturing 1 year after

• Or he simultaneously borrows and lends $10,000 repayable at the end of year 2 and year 1, respectively

• This is equivalent to borrowing $10,000x[1+R(0,1)] in one year, repayable in two years as $10,000x(1+R(0,2))2.

– The implied rate on the loan given by the following equation is the forward rate F(0,1,1)

1,01

2,011,1,0

2

R

RF

Futures Pricing

• Price futures contracts by using replication argument, just like for forward contracts

• Let’s consider two otherwise identical forward and futures contracts

– Cash-flows are not identical because gains and losses in futures trading are paid out at the end of the day

– Denoted as G0 and F0, respectively, current forward and futures prices

• When interest rates are changing randomly– Cannot create a replicating portfolio– Cannot price futures contracts by arbitrage

• However, short term bond prices are very insensitive to interest rate movements

– Replication argument is almost exact

Futures versus Forward Pricing

Date Forward Contract Futures Contract

0 0 0

1 0 F1 F0

2 0 F2 F1

3 0 F3 F2

. . .

. . .

. . .

T 1 0 FT 1 FT 2

T PT G0 PT FT 1

Total PT G0 PT F0

Uses of Futures

• Fixing today the financial conditions of a loan or investment in the future

• Hedging interest rate risk – Because of high liquidity and low cost due to low margin

requirements, futures contracts are actually very often used in practice for hedging purposes

– Can be used for duration hedging or more complex hedging strategies (see Chapters 5 and 6)

• Pure speculation with leverage effect– Like bonds, futures contracts move in the opposite direction to

interest rates– This is why a speculator expecting a fall (rise) in interest rates will

buy (sell) futures contracts– Advantages : leverage, low cost, easy to sell short

Uses of Futures – Con’t

• Detecting riskless arbitrage opportunities using futures

• Cash-and-carry arbitrage – Consists in buying the underlying asset and selling the forward or

futures contract– Amounts to lending cash at a certain interest rate X– There is an arbitrage opportunity when the financing cost on the

market is inferior to the lending rate X

• Reverse cash-and-carry – Consists in selling (short) the underlying asset and buying the

forward or futures contract– Amounts to borrowing cash at a certain interest rate Y– There is an arbitrage opportunity when the investment rate on the

market is superior to the borrowing rate Y