Chapter 11 Forwards and Futures

FIXED-INCOME SECURITIES Chapter 11 Forwards and Futures

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)

Instruments

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

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)

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$ 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

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 -

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 • or • or • 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

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)

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

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

Chapter 11 Forwards and Futures