Principles of Futures Contract Pricing

The expectations hypothesis Normal backwardation A full carrying charge market Reconciling the three theories

The Expectations Hypothesis

The expectations hypothesis states that the futures price for a commodity is what the marketplace expects the cash price to be when the delivery month arrives– Price discovery is an important function

performed by futures

There is considerable evidence that the expectations hypothesis is a good predictor

Normal Backwardation

Basis is the difference between the future price of a commodity and the current cash price– Normally, the futures price exceeds the cash

price (contango market)– The futures price may be less than the cash

price (backwardation or inverted market)

Normal Backwardation (cont’d)

John Maynard Keynes:– Locking in a future price that is acceptable

eliminates price risk for the hedger– The speculator must be rewarded for taking the

risk that the hedger was unwilling to bear Thus, at delivery, the cash price will likely be

somewhat higher than the price predicated by the futures market

A Full Carrying Charge Market

A full carrying charge market occurs when the futures price reflects the cost of storing and financing the commodity until the delivery month

The futures price is equal to the current spot price plus the carrying charge:

CSF t

A Full Carrying Charge Market (cont’d)

Arbitrage exists if someone can buy a commodity, store it at a known cost, and get someone to promise to buy it later at a price that exceeds the cost of storage

In a full carrying charge market, the basis cannot weaken because that would produce an arbitrage situation

Reconciling the Three Theories The expectations hypothesis says that a futures

price is simply the expected cash price at the delivery date of the futures contract

People know about storage costs and other costs of carry (insurance, interest, etc.) and we would not expect these costs to surprise the market

Because the hedger is really obtaining price insurance with futures, it is logical that there be some cost to the insurance

Spreading with Futures

Intercommodity spreads Intracommodity spreads Why spread in the first place?

Intercommodity Spreads

An intercommodity spread is a long and short position in two related commodities

– E.g., a speculator might feel that the price of corn is too low relative to the price of live cattle

– Risky because there is no assurance that your hunch will be correct

With an intermarket spread, a speculator takes opposite positions in two different markets

– E.g., trades on both the Chicago Board of Trade and on the Kansas City Board of Trade

Intracommodity Spreads

An intracommodity spread (intermonth spread) involves taking different positions in different delivery months, but in the same commodity– E.g., a speculator bullish on what might buy

September and sell December

Why Spread in the First Place?

Most intracommodity spreads are basis plays Intercommodity spreads are closer to two

separate speculative positions than to a spread in the stock option sense

Intermarket spreads are really arbitrage plays based on discrepancies in transportation costs or other administrative costs

Pricing of Stock Index Futures

Elements affecting the price of a futures contract Determining the fair value of a futures contract Synthetic index portfolios

Elements Affecting the Price of A Futures Contract

The S&P 500 futures value depends on four elements:– The level of the spot index – The dividend yield on the 500 stock in the index– The current level of interest rates– The time until final contract cash settlement

Elements Affecting the Price of A Futures Contract (cont’d)

S&P 500 Stock Index

Futures

SPX Index

T-bill Rate Time until Settlement

SPX Dividend Yield

Elements Affecting the Price of A Futures Contract (cont’d)

Stocks pay dividends, while futures do not pay dividends

– Shows up as a price differential in the futures price/underlying asset relationship

Stocks do not accrue interest

Posting margin for futures results in interest– Shows up as a price differential in the futures

price/underlying asset relationship

Determining the Fair Value of A Futures Contract

The futures price should equal the index plus a differential based on the short-term interest rate minus the dividend yield:

TDRSeF )(

Determining the Fair Value of A Futures Contract (cont’d)

Calculating the Fair Value of A Futures Contract Example

Assume the following information for an S&P 500 futures contract:

Current level of the cash index (S) = 1,484.43 T-bill yield ® = 6.07% S&P 500 dividend yield (D) = 1.10% Days until December settlement (T) = 121 = 0.33 years

Determining the Fair Value of A Futures Contract (cont’d)

Calculating the Fair Value of A Futures Contract Example

The fair value of the S&P 500 futures contract is:

30.509,143.484,1 )365/121)(0110.0607(.

)(

e

SeF TDR

Synthetic Index Portfolios Large institutional investors can replicate a well-

diversified portfolio of common stock by holding– A long position in the stock index futures contract and– Satisfying the margin requirement with T-bills

The resulting portfolio is a synthetic index portfolio The futures approach has the following advantages over

the purchase of individual stocks:– Transaction costs will be much lower on the futures contracts– The portfolio will be much easier to follow and manage

Basic Convergence: As time passes, the difference between the cash index and the futures price will narrow

– At the end of the futures contract, the futures price will equal the index (basic convergence)

Interest Rate Futures Exist across the yield curve and on many

different types of interest rates– T-bond contracts– Eurodollar (ED) futures contracts– 30-day Federal funds contracts– Other Treasury contracts

Characteristics of U.S. Treasury Bills

Sell at a discount from par using a 360-day year and twelve 30-day months

91-day (13-week) and 182-day (26-week) T-bills are sold at a weekly auction

Characteristics of U.S. Treasury Bills (cont’d)

Treasury Bill Auction ResultsTerm Issue Date Auction

DateDiscount Rate %

Investment Rate %

Price Per $100

13-week 01-02-2004 12-29-2003 0.885 0.901 99.779

26-week 01-02-2004 12-29-2003 0.995 1.016 99.500

4-week 12-26-2003 12-23-2003 0.870 0.882 99.935

13-week 12-26-2003 12-22-2003 0.870 0.884 99.783

26-week 12-26-2003 12-22-2003 0.970 0.992 99.512

4-week 12-18-2003 12-16-2003 0.830 0.850 99.935

Characteristics of U.S. Treasury Bills (cont’d)

The “Discount Rate %” is the discount yield, calculated as:

Days

360

ValuePar

PriceMarket - ValuePar YieldDiscount

Characteristics of U.S. Treasury Bills (cont’d)

Discount Yield Computation Example

For the first T-bill in the table on slide 6, the discount yield is:

%884.090

360

000,10

90.977,9000,10

Days

360

ValuePar

PriceMarket - ValuePar YieldDiscount

Characteristics of U.S. Treasury Bills (cont’d)

The discount yield relates the income to the par value rather than to the price paid and uses a 360-day year rather than a 365-day year– Calculate the “Investment Rate %” (bond

equivalent yield):

maturity toDays

365

PriceDiscount

AmountDiscount Yield Equivalent Bond

Characteristics of U.S. Treasury Bills (cont’d)

Bond Equivalent Yield Computation Example

For the first T-bill in the table on slide 6, the bond equivalent yield is:

%90.090

365

90.977,9

90.977,9000,10

maturity toDays

365

PriceDiscount

AmountDiscount Yield Equivalent Bond

The Treasury Bill Futures Contract

Treasury bill futures contracts call for the delivery of $1 million par value of 91-day T-bills on the delivery date of the futures contract– On the day the Treasury bills are delivered, they

mature in 91 days

The Treasury Bill Futures Contract (cont’d)

Futures position 91-day T-bill T-bill

established delivered matures

91 days

Time

The Treasury Bill Futures Contract (cont’d)

T-Bill Futures Quotations

September 15, 2000

  Open High Low Settle Change Settle Change Open Interest

Sept 94.03 94.03 94.02 94.02 -.01 5.98 +.01 1,311

Dec 94.00 94.00 93.96 93.97 -.02 6.03 +.02 1,083

Characteristics of Eurodollars

Applies to any U.S. dollar deposited in a commercial bank outside the jurisdiction of the U.S. Federal Reserve Board

Banks may prefer eurodollar deposits to domestic deposits because:– They are not subject to reserve requirement

restrictions– Every ED received by a bank can be reinvested

somewhere else

The Eurodollar Futures Contract

The underlying asset with a eurodollar futures contract is a three-month, $1 million face value instrument– A non-transferable time deposit rather than a

security The ED futures contract is cash settled with no actual

delivery

The Eurodollar Futures Contract (cont’d)

Treasury Bill vs Eurodollar FuturesTreasury Bills Eurodollars

Deliverable underlying commodity Undeliverable underlying commodity

Settled by delivery Settled by cash

Transferable Non-transferable

Yield quoted on discount basis Yield quoted on add-on basis

Maturities out to one year Maturities out to 10 years

One tick is $25 One tick is $25

The Eurodollar Futures Contract (cont’d)

The quoted yield with eurodollars is an add-on yield

For a given discount, the add-on yield will exceed the corresponding discount yield:

Maturity toDays

360

icePr

DiscountYieldon -Add

The Eurodollar Futures Contract (cont’d)

Add-On Yield Computation Example

An add-on yield of 1.24% corresponds to a discount of $3,124.66:

$3,124.66Discount

91

360

Discount000,000,1$

Discount0124.

Maturity toDays

360

icePr

DiscountYieldon -Add

The Eurodollar Futures Contract (cont’d)

Add-On Yield Computation Example (cont’d)

If a $1 million Treasury bill sold for a discount of $3,124.66 we would determine a discount yield of 1.236%:

%236.191

360

$1,000,000

$3,124.66 YieldDiscount

Speculating With Eurodollar Futures

The price of a fixed income security moves inversely with market interest rates

Industry practice is to compute futures price changes by using 90 days until expiration

Speculating With Eurodollar Futures (cont’d)

Speculation Example

Assume a speculator purchased a MAR 05 ED futures contract at a price of 97.26. The ED futures contract has a face value of $1 million. Suppose the discount yield at the time of purchase was 2.74%. In the middle of March 2005, interest rates have risen to 7.00%.

What is the speculator’s dollar gain or loss?

Speculating With Eurodollar Futures (cont’d)

Speculation Example (cont’d)

The initial price is:

150,993$360

900274.1000,000,1$Price

360

90YieldDiscount -1Value FacePrice

Speculating With Eurodollar Futures (cont’d)

Speculation Example (cont’d)

The price with the new interest rate of 7.00% is:

00.500,982$360

900700.1000,000,1$Price

360

90YieldDiscount -1Value FacePrice

Speculating With Eurodollar Futures (cont’d)

Speculation Example (cont’d)

The speculator’s dollar loss is therefore:

00.650,10$00.150,993$00.500,982$

Hedging With Eurodollar Futures

Using the futures market, hedgers can lock in the current interest rate

Hedging With Eurodollar Futures (cont’d)

Hedging Example

Assume you are a portfolio managers for a university’s endowment fund which will receive $10 million in 3 months. You would like to invest the money now, as you think interest rates are going to decline. Because you want a money market investment, you establish a long hedge in eurodollar futures. Using the figures from the earlier example, you are promising to pay $993,150.00 for $1 million in eurodollars if you buy a futures contract at 98.76. Using the $10 million figure, you decide to buy 10 MAR ED futures, promising to pay $9,969,000.

Hedging With Eurodollar Futures (cont’d)

Hedging Example (cont’d)When you receive the $10 million in three months, assume interest rate have fallen to 1.00%. $10 million in T-bills would then cost:

This is $6,000 more than the price at the time you established the hedge.

00.000,975,9$360

9001.1000,000,10$Price

Hedging With Eurodollar Futures (cont’d)

Hedging Example (cont’d)

In the futures market, you have a gain that will offset the increased purchase price. When you close out the futures positions, you will sell your contracts for $6,000 more than you paid for them.

Treasury Bonds and Their Futures Contracts

Characteristics of U.S. Treasury bonds Pricing of Treasury bonds The Treasury bond futures contract Dealing with coupon differences The matter of accrued interest Delivery procedures The invoice price Cheapest to deliver

Characteristics of U.S. Treasury Bonds

Very similar to corporate bonds:– Pay semiannual interest– Have a maturity of up to 30 years– Are readily traded in the capital markets

Different from Treasury notes:– Notes have a life of less than ten years– Some T-bonds may be callable fifteen years

after issuance

Characteristics of U.S. Treasury Bonds (cont’d)

Bonds are identified by:– The issuer– The coupon– The year of maturity

E.g., “U.S. government six and a quarters of 23” means Treasury bonds with a 6¼% coupon rate that mature in 2023

Dealing With Coupon Differences

To standardize the $100,000 face value T-bond contract traded on the Chicago Board of Trade, a conversion factor is used to convert all deliverable bonds to bonds yielding 6%

Dealing With Coupon Differences (cont’d)

N whole theof excessin months ofnumber theX

maturity toyears wholeofnumber N

form decimalin coupon annualC

factor conversion CF

where

6

X6

2)03.1(

1

)03.1(

11

06.02(1.03)

1 CF

2N2N6

x

CCC

Cheapest to Deliver

Normally, only one bond eligible for delivery will be cheapest to deliver

A hedger will collect information on all the deliverable bonds and select the one most advantageous to deliver