FIXED-INCOME SECURITIES
Chapter 11
Forwards and Futures
Outline
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•
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•
•
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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 agreedupon 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
– Short : believe price will fall
• 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
• Le Marché à Terme International de France (www.matif.fr)
• Eurex (www.eurexchange.com)
Instruments
CME
Eurodollar Futures
13-Week Treasury Bill Futures
CBOT
30-Year US Treasury Bonds
10-Year US Treasury Notes
Libor Futures
5-Year US Treasury Notes
Fed Funds Turn Futures
10-Year Agency Futures
5-Year Agency Futures
Argentine 2X FRB Brady Bond
Futures
Argentine Par Bond Futures
2-Year US Treasury Notes
10-Year Agency Notes
5-Year Agency Notes
Long Term Municipal Bond
Index
30-Day Federal Funds Mortgage
Brazilian 2 X C Brady Bond
Futures
Brazilian 2 X EI Brady Bond
Futures
Mexican 2 X Brady Bond
Futures
Euro Yen Futures
Japanese Government Bond
Futures
Euro Yen Libor Futures
Mexican TIIE Futures
Mexican CETES Futures
LIFFE
Long Gilt Contract
German Government Bond
Contract
Japanese Government Bond
Contract
3-Month Euribor Future
3-Month Euro Libor Future
3-Month Sterling Future
3-Month Euro Swiss Franc
Future
3-Month Euroyen (TIBOR)
Future
3-Month Euroyen (LIBOR)
Future
2-Year Euro Swapnote
5-Year Euro Swapnote
10-Year Euro Swapnote
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 no 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 10year 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
10
50
1,000
CF  

$926.3991
10
i
1  6%
i 1 (1  6%)
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 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 = $100,000
– Price= 97
• Three bonds denoted A,B and C
Bond A
Bond B
Bond C
Quoted Price
Conversion Factor
IP-CP
103.90
118.90
131.25
107.145%
122.512%
135.355%
3,065$
-6,336$
4,435$
• 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
– 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
Buy 1 unit of bond B
Tt
360
Pt AC 100 c 
Tt
365
Forward Pricing
• Given that both trades have a cost equal to zero at
date t, in the absence of arbitrage opportunities, the
cash-flows generated by the two operations at date T
must be equal
• From this we obtain

 T  t 
T t 
Ft  Pt  1  r 
  100  c  

 360 
 365 

or

 T  t 
Ft  Pt  1  R  C 

 365 

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

 184 
 184 
F05/ 01/ 01  115  1  7%

100

10
%



  114.07
 360 
 365 

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)
2
F 0,1,1 
1  R0,2
1  R0,1
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
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 G 0
PT FT1
Total
PT G 0
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