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Subject 5B: Workshop 1 PART 1: FORWARDS AND FUTURES

(reading: Hull 6th

or 7th

edition chapters 1-3 and 5)

A financial derivative is a financial instrument whose value depends on oris derived from the values of other financial instruments or other more

basic underlying variables (such as commodity prices, exchange rates,share prices, interest rates etc). These contracts range from very simple tovery complex.

Exchange Traded Markets

A derivative exchange is a market where individuals trade in standardizedcontracts that have been defined by the exchange. The CBOT wasestablished in 1848 to bring farmers and merchants together and initially

its task was to standardize the quantities and qualities of the grains tradedthere. Futures type contracts were developed within a few years andspeculators soon became interested in it. Many market participants foundtrading the futures contract to be a good alternative to trading the grainitself. Most exchanges have organized arrangements so as to almosteliminate counterparty credit risk from trading. This is the risk that the

person you are dealing with defaults on their obligations.

Over the counter markets

Not all trading is done on exchanges such as the ASX, SFE, CBOT. Asignificant amount of trading is done directly between buyer and sellerinstead of via an exchange. With trades on an exchange there is usually anintermediary involved (e.g. a broker) and both buyer and seller deal witheach other via the intermediary. With OTC markets the deal is donedirectly between the 2 parties. Another feature of the OTC markets is thatthe contract details (quantity and quality and other features) are notstandardized but are individually negotiated. However the degree of

counterparty credit risk is higher.

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Spot vs Forward contracts

A spot contract is an agreement to buy or sell an asset today: the exchangeof money for the instrument concerned happens immediately (the sameday).

A forward contract is an agreement to buy or sell some asset at a certainfuture time at a certain future price. These are very common in the foreignexchange market.

Broker vs. Market Maker

A broker is a market participant who acts as an intermediary to atransaction. The role of broker is to match up the buyer and the seller.

They charge both parties a fee for this service. In the share market, themarket arrangements require investors to use the services of brokers to buyor sell their shares. You have to use a broker whether you want to or not.The broker does not own the stock themselves, and they are not exposed torisk of the price of the stock changing. The same applies in the futuresexchanges (e.g. the SFE) and options exchanges: you have to deal via a

broker.

A real estate agent is the equivalent of a broker for the real estate market.

They match buyer and seller and charge a fee to the seller for selling theproperty. The seller pays for it out of the price paid by the buyer. Theagent does not actually have ownership of the property being sold.

A Market Maker is another type of market participant who gets involvedin transactions in securities. The difference is that a market maker actuallyowns the security being traded and is exposed to the risk that prices movein an unfavourable direction.

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Bid and Offer Prices

A market maker is someone who offers to both buy and sell the securityconcerned. They will quote a 2 way price to the market. The bid price isthe price at which they offer to buy the stock from you. The offer price

is the price at which they are willing to sell the stock to you. The offerprice is always higher than the bid price. The difference between the offerprice and the bid price is called the bid offer spread.

The size of the bid offer spread reflects the transaction costs involved forthe dealer, as well as the liquidity of the market. A large bid offer spreadindicates either a very illiquid market or a high level of risk for the dealeror a lack of competition. A low bid offer spread indicates a very active andliquid market with lots of competition.

Banks are market makers in the foreign exchange market

Example: table 1.1 spot and forward quotes for the USD-GBP exchangerate

bid offer

spot 1.4452 1.4456

1-month forward 1.4435 1.4440

3-month forward 1.4402 1.44076-month forward 1.4353 1.4359

1-year forward 1.4262 1.4268

This table shows the quotes on the exchange rate of the british pound(GBP) for the US dollar (USD) by a large international bank on 16 aug2001. The quote means the number of units of the USD for 1 unit of theGBP.

The bid rate means the number of USD the bank is prepared to pay to buy1 GBP.

The offer rate means the number of USD the bank is prepared to sell 1GBP for.

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Note that the offer rate is above the bid rate (otherwise the bank wouldlose money and banks are in the business of making money not oflosing it).

The spot bid quote means that the bank will buy 1 GBP from thecustomer in exchange for 1.4452 US dollars for immediate delivery.(buy GBP & Sell USD).

The spot ask quote means that the bank is willing to sell GBP to thecustomer in exchange for 1.4456 USD (sell GBP / but USD) forimmediate delivery.

The 6-month forward exchange rate quotes mean that

the bank is willing to enter a deal to buy 1 GBP for 1.4353 USD in sixmonths time (both amounts to be exchanged in the future, not now)

the bank is willing to enter a deal to sell 1GBP for 1.4359 USD in sixmonths time.

Forward Contracts

The simplest type of financial derivative contract is the forward contract.A forward contract is an agreement (betweeen 2 parties) made today to

purchase some instrument on some future date for a price that is agreedupon today. Note that a forward contract is an over the countertransaction (not exchange traded).

The date on which the exchange of money in return for the instrument iscalled the delivery date or the maturity date or the expiry date. T isour notation for the delivery date.

The asset which is being bought is called the underlying asset orsometimes just the underlying. We will use S to denote the value of theunderlying asset and the subscript t to denote the time. ST is the value ofthe underlying asset on the maturity date of the forward contract.

The amount of money which will change hands on the delivery date iscalled the delivery price which we will denote by X

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When the forward contract is entered into the delivery price is set at a levelsuch that there is no need to have any exchange of money between the 2

parties. The only exchange of money occurs on the delivery date.

One party to the contract is obligated to buy the underlying asset on the

maturity date and to pay an amount X for it. This applies regardless of themarket value of that asset at the time. The other party is obligated to sellthe asset on the maturity date in return for a payment of X.

Both sides are locked into the deal. It is a zero sum game and if thecontract is making a profit for one party it is making a loss of the samemagnitude for the other party.

This means if you have a forward contract in place which is of positive

value, it is of negative value for the other party (called the counterparty.The counterparty could default on their obligations under the contract, andthis is one of the risks of forward contracts.

The party who is obligated to buy the asset is said to have aLONG POSITION

The party who is obligated to sell the asset is said to have aSHORT POSITION

The payoff to the party with the long position is, on the maturity date of

the contract, ( )Tpayoff S X= .

The payoff to the holder of the short position is 1 times this, which is

( )Tpayoff X S= .

The value of the asset on the maturity date cannot be known with certainty,

it is a random variable.

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Investment vs Consumption Assets

Investment Asset: one that is held for investment purposes by a significantnumber of investors e.g. bonds, shares, gold, silver

consumption asset: one that is held primarily for consumption: e.g. copper,oil, pork bellies (bacon)

We can use arbitrage type arguments to produce valuation formulae forforward and futures contracts over investment assets. This is not possiblefor forwards and futures defined over consumption assets.

Short Selling:

An important concept in the valuation of options and forward and futures

contracts is that of short selling of an asset. Short selling means sellingsomething you do not own. This often causes confusion how can you sellsomething you do not own? For most physical assets we use in our dailylives you cannot do this. If you sell your neighbours car it would beillegal. But with financial assets you can.

The way it works is this

you borrow the asset (say NAB shares) from another investor (often abroker) who does own them, but you promise to give it back at a later date

you then sell them in the market for the then current market price you use the sale proceeds for whatever purpose you had in mind later on you have to buy the asset back in the market for the price

prevailing on the day you buy it back

you then give it back to the party you borrowed it from you may have to pay a fee and compensation for any income on the asset

during the term of the arrangement

Doing this will deliver a profit to the investor if the price of the asset fallsbut deliver a loss if the price of the asset increases. Short selling can anddoes happen in financial markets. There are some restrictions on shortselling but we will assume there are no such restrictions.

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Types of market participant:

Hedgers: traders who want to protect themselves from financial loss byentering into a transaction. For example, if you are an importer, you areexposed to the risk of the local currency falling in value relative to the

currency of the country you import goods from. You can protect yourselfagainst this risk by entering into a forward foreign exchange transaction.

Speculators: traders who want to make short term profits by entering intoa transaction, by taking a bet on which way market prices are going tochange. For example, you may have the opinion that defence related stocksare going to go up in value next week due to a new war breaking out incentral asia this is likely to increase demand for the goods and services

produced by the defence industry. If you buy now and sell next week you

may make a substantial profit (assuming your guess about the war wascorrect).

Arbitrageurs: traders who engage in arbitrage. This means making aseries of transactions simultaneously in 2 or more markets in such a way asto make a riskless profit. For instance some stocks are traded in both NewYork and London. If a particular stock is trading for USD152 in New Yorkand for GBP100 in London at a time when the exchange rate is 1GBP =1.55USD, then a trader could do the following 2 deals:

(i) buy 10000 shares in New York for $US1,520,000(ii) sell 10000 shares in London for $US1,550,000 = GBP 1,000,000

1.55(iii) get a risk free profit of $30,000 in USDIn practice there are transactions costs involved in doing this: there aretransactions costs involved in buying the stock, in selling the stock and inconverting one currency to another. For small investors these transactioncosts tend to be large ( as a proportion of the profits) but for big investorssuch as banks they are much lower.

The actions of investors to buy the cheap version (in New York) and sellthe expensive version (in London) of the same asset will tend to make the2 prices converge: the cheap one gets more expensive and the expensiveone gets cheaper until there is no profit to be made from arbitrage.

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Relationship between spot and forward prices

Notation and assumptions

no transaction costs market participants have same tax rate all market participants can borrow or lend any amount of money at the

same risk free interest rate (for both borrowing and lending) market participants can exploit arbitrage opportunities as they occur these assumptions apply (at least approximately) to a few key market

participants such as large investment banks

T = time until delivery (maturity) in a forward / futures contract 0S = price of the underlying asset today 0F = futures price today for a futures contract maturing at time T r = risk free interest rate (usually taken to be libor)Forward Price for an investment asset

The easiest type of forward contract to value is one where the underlyingasset pays no income during the term of the contract: zero coupon bondsand non dividend paying stocks are examples of this type of asset.

Suppose that a forward contract that matures at time T over the asset S hasdelivery price K. How should the delivery price K be chosen?

Suppose that the underlying asset is a stock with initial price 0 40S = and

the term of the contract is 0.25T = years (i.e. 3 months), and that the riskfree interest rate is 0.05r = continuously compounded.

Q: What would happen if the delivery price were K = 43?Under this scenario, an arbitrageur could

Now:

borrow $40 for 3 months at 5% buy the stock for $40 short the forward contract (agree to sell the stock at time 3 months for a

price of $43Wait till the end of 3 months:

sell the stock and receive $43, pay back the loan with interest for 0.05 0.2540.50 40 e = keep the profit of $2.50

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What would happen if the delivery price were K = $39?

Under this scenario, an arbitrageur couldNow:

short sell the stock for $40 now, agreeing to give it back to the personthe stock was borrowed from (by purchasing it in the open market in 3months time for its price at that time)

invest the $40 for 3 months at 5% (i.e. lend risk free for 3 months) go long in the forward contract (agree to buy the stock at time 3 months

for a price of $39)wait till the end of 3 months:

buy the stock and pay $39, give back the stock to the person you borrowed it from under the short

selling arrangement Receive the proceeds of our risk free investment which is

0.05 0.2540.50 40 e = keep the profit of $1.50 = 40.50 - 39.00The only delivery price which does not permit an arbitrageur to make a

profit from doing one of the above strategies is K = $40.50. This value ofK is denoted by F and is called the arbitrage free forward price.

The mathematical formula for what K should be is:

0 0.rT

F S e=

Short selling: what if we cant do this?

Actually it does not matter if short selling cant be done (e.g. because it isillegal it used to be illegal in australia).

If it is an investment asset then a lot of people hold the asset forinvestment purposes and are willing to sell if the price is right. If theforward price is too low, then an arbitrageur (who holds the asset) can sellthe asset and go long in a forward contract.

0 0.rT

F S e= is the arbitrage free forward price

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If the delivery price K for a forward contract is 0K F> then the

arbitrageurs strategy is:

Now

borrow 0S for term T at rate rbuy the asset and pay 0S for it

enter a short forward contract to sell S at time T for amount K

Intermediate

hold this position to time TAt time T

pay back loan with interest for amount 0 0.rT

F S e= sell the asset and get K for it profit at time T = 0 0K F >

The combination of borrowing and buying the asset as above, creates what iscalled a synthetic long forward position in the asset.

The payoff at maturity from creating this synthetic long forward contract is

( )1 0Tpayoff S F=

The payoff from the short forward contract with delivery price K is

( )2 Tpayoff K S=

The overall payoff is the sum of these 2 payoffs: this is

( ) ( )0 0T Tpayoff S F K S K F= + =

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If the delivery price K is 0K F< then the strategy

(for a holder of the asset) is:

Now

sell the asset for amount 0S invest / lend the proceeds 0S for term T at rate r enter a long forward contract to buy the asset at time T for amount KIntermediate

hold this position to time TAt time T

loan matures with interest for amount 0 0. rTF S e= buy the asset and pay K for it payoff at time T = 0TS K F +

Alternatively the investor who owns the asset could have decided not tosell the asset and instead hold it and wait till time T. In this case the payoff

at time T (value of the position) would be justT

S

The profit to be had from deciding to adopt the sell / lend / long forwardstrategy instead of deciding to hold is ( ) ( )0 0T TS K F S F K + = .If this is positive we prefer to follow the above strategy instead of holdingonto the asset.

The combination of selling the asset and lending the sale proceeds at therisk free rate for term T creates what is called a synthetic short forward

position in the asset. The payoff at maturity from doing this is

( )1 0 Tpayoff F S=

The payoff from the long forward contract with delivery price K is

( )2 Tpayoff S K=

The overall payoff is the sum of these 2 payoffs: this is

( ) ( )0 0T Tpayoff F S S K F K= + =

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Forward Contract when the asset provides a known income

Suppose the underlying asset provides a perfectly predictable cash incometo the holder during the term of the forward contract. In this case the

formula for the arbitrage free forward price is

( )0 0 .rT

F S I e=

where I is the present value (at the risk free rate) of the income producedby the asset during the term of the forward contract.

Numerical Example:

we have a 10 month forward contract on a stock with initial price 0 50S = the risk free interest rate is 8% pa continuously compounded the stock pays a dividend of $0.75 every 3 months, the first due in exactly

3 months.

The present value of the income paid during the term of the forward

contract is ( )0.08 0.25 0.08 0.50 0.08 0.750.75 2.162I e e e = + + = The forward price is

( ) ( )10

0.0812

0 0 . 50.000 2.162 51.13584rT

F S I e e

= = =

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If the delivery price K is 0K F> then the strategy is:

Now

borrow 0S for term T at rate r buy the asset and pay 0S for it the income stream provided by the asset has PV of I enter a short forward contract to sell S at time T for amount KIntermediate

hold this position to time T reinvest any income as it is received at the risk free rate of interestAt time T

the reinvested income will have accumulated to an amount . rTI e pay back loan with interest for amount 0. rTS e sell the asset and get K for it profit at time T =

NN

N( )0 0 0

'

0rT rT rT

sale accumulatedloanproceeds reinvestedmaturity

incomepayment

K S e Ie K S I e K F + = = >

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If the delivery price K is 0K F< then the strategy for a holder of the asset

is (if short selling is not allowed):

Now:

sell the asset and get 0S for it lend the proceeds of 0S for term T at rate r enter a long forward contract to buy S at time T for amount KIntermediate: hold this position to time T

At time T:

the loan matures and we receive amount 0. rTS e buy the asset and pay K for it payoff at time T =N N0

'

rTT

valueloanofmaturityassetpayment

S e K S +

Alternatively the investor could have held on to the asset instead of sellingit. The income provided by the asset could have been reinvested at the risk

free interest rate up to time T.

The time T payoff provided by this alternative strategy would have been

NN. rT

T

accumulatedvaluereinvestedatincomematurity

S I e+

The extra benefit provided by selling the asset, lending the proceeds andgoing long a forward, instead of just holding the asset, is the difference

between these 2 payoffs: this is( ) ( ) ( )0 0 0. 0rT rT rT T TS e K S S I e S I e K F K + + = = >

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if short selling is allowed and if 0K F< then the strategy to make an

arbitrage profit is:

Now

short sell the asset and get 0S for it (this involves borrowing theasset and compensating the lender for any income foregone duringthe term of the short selling arrangement)

lend the proceeds of 0S for term T at rate r enter a long forward contract to sell S at time T for amount KIntermediate: hold this position to time T

At time T the loan matures and we receive amount 0. rTS e buy the asset and pay K for it to the counterparty of the forward

contract

give the asset (worth TS ) back to the person you borrowed it fromunder the short selling arrangement

pay compensation (to the person who lent you the asset for theshort selling) for the income foregone of amount

N. rT

accumulatedreinvestedincome

I e

payoff =N

N( )0 0 0

'

. 0rT rT rT

accumulatedloanreinvestedmaturityincomepayment

S e K I e S I e K F K = = >

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Forward Contract when the asset provides a known income yield

By income yield we mean the ratio of the income to the value of the asset.Income yields are usually taken to mean continuous income yields. Thisis analogous to continuously compounded interest rates.

Suppose the underlying asset provides a constant income yield at rate yp.a. to the holder during the term of the forward contract. This means that

during a small time interval ( ),t t t+ , the amount of income paid on theasset is . .

t ty t S+ , which is proportional to the asset value at the time of

payment.

An example of an asset that behaves like this is a holding of a foreigncurrency in a foreign bank account which pays interest at a fixed rate buton a daily basis. Suppose you have USD1m in a US bank account which

pays interest at 6% p.a. on a daily basis. The interest earned is added to theaccount balance every day. When converted into AUD, the interest incomeis proportional to the amount of the account.

In this case the formula for the arbitrage free forward price is given by the

formula( )

0 0.r y T

F S e=

In deriving this forward price we make the assumption that the incomereceived on the asset can be reinvested back into the asset. For instance inthe case of shares, the dividends could be used to purchase more shares.

If we hold one unit of the asset at time t=0, at a price of 0S and if we

reinvest any income back into more units of the asset, then by time T we

will hold yTe units of the asset and these are worthT

S per unit.

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Numerical Example:

Problem

The dividend yield on an asset is 4% p.a. convertible half yearly, and therisk free rate is 10% p.a. with continuous compounding. The asset price isinitially $25. What is the forward price for a forward contract maturing in6 months?

Solution: First we need to convert the dividend yield to an equivalent

continuous yield. This is 2ln(1.02 ) 2 ln(1.02) 0.0396y = = =

From the information given we have

( ) ( )

0

10.10 0.0396

20 0

25.00

0.50

0.10

0.0396

25 $25.77r t T

S

T

r

y

F S e e

=

=

=

=

= = =

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Valuation of forward contracts:

It is normal market practice with forward contracts to set the forward priceto be the arbitrage free forward price when the contract is first set up. Thismeans that when the contract is initiated, it is designed so that there is no

exchange of cash for the asset, and the exchange happens only at thematurity date. As at the inception date, the value of the contract is zero to

both sides of the deal.

Later on, after the contract has been set up, market conditions willprobably change 0from what they were initially. Consequently the value ofthe contract will also change and it is unlikely to stay at zero. We shallconsider the value of the contract from the perspective of the holder of thelong position. This value can change from positive to negative and back

again during the life of the contract.

Consider a forward contract over some asset S, written at some previoustime

let K be the delivery price of the forward contract

Let the term to maturity be T

Let 0F be the forward price of the asset S for delivery at time T for anew forward contract written today

Let fbe the market value today of the forward contract which

matures at time T with delivery price K

Then the market value of the forward contract is ( )0rT

f F K e=

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Proof:

If we have a long position in a forward contract with maturity T andexercise price Kthen

we can at zero cost, also enter into a short forward contract withmaturity T and delivery price 0F This combination of forward contracts has a payoff which is risk free

and is independent of the stock price at maturity,TS

The payoff from this combination is( ) ( )0 0T Tpayoff S K F S F K= + =

The present value of this payoff is ( )0 rTF K e The present value of the payoff( )0 TF S is zero by design Hence the present value of the payoff( )TS K is ( )0 rTF K e

Note:(i) for an investment asset that pays no income during the term of the

forward contract we have 0 0rT

F S e= so that the value of a forward

contract with delivery price K is

( ) ( )0 0 0rT rT rT rT f F K e S e K e S Ke = = =

(ii) if ( )0 0 rTF S I e= then 0 rTf S I Ke= (iii) if ( )0 0 r y TF S e = then 0 yT rTf S e Ke = Numerical example of valuation of a forward contract:

Scenario: Six months ago you entered into a long forward contract for aterm of 12 months. Today this forward contract has 6 months remaining tomaturity.

The delivery price of the contract is X = $24The current price of the underlying asset is S = 25The risk free rate of interest is 10% p.a. with continuous compoundingThe asset pays no income during the remaining term of the contract

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What is the forward price of the asset for a forward contract written todaythat matures in 6 months?

What is the value of the forward contract with delivery price of $24? What would the value of the contract be if it was a short forward contract

instead? Assuming that interest rates 6 months ago were 10% and that $24 was the

arbitrage free forward price at that time for a 1 year contract, what was thevalue of the underlying asset then?

Solution

The forward price of the asset is 0.10 0.50 0 $25.00 26.28rTF S e e = = =

If a new forward contract on the asset were written today, this is what thedelivery price would be, and it would be costless to enter into this contractas either the holder of a long or the holder of a short position.

To compute the value today of the long forward contract that waswritten 6 months ago with a delivery price of $24, the valuation

formula is ( ) ( ) 0.12 0.50 26.28 24 2.17rT

f F K e e = = =

This is the present value of the payoff that would apply if the price of

the underlying asset on the maturity date is equal to the forward price.

If the contract were a short forward instead of a long forward then thevalue of the contract would be of the same magnitude but opposite

sign: ( ) ( ) 0.12 0.50 24 26.28 2.17rT

f K F e e = = =

To back solve for what the share price would have been 6 months agowe solve for S and get

0.10 0.10

0 0 $24.00 24.00 21.72

rT

F S e S e S e

= = = =

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Equality of futures and forward prices:

When the risk free interest rate is constant and the same for all maturities,then it can be shown that the forward price and the futures price for acontract with a certain delivery date are the same. The proof (see

textbook) can be extended to the situation where the interest rate is notconstant but is known with certainty.

In the real world, interest rates vary unpredictably, and in thesecircumstances forward and futures prices are not the same.

As explained in the textbook: if the asset price is highly positivelycorrelated with interest rates then futures prices tend to be higher thanforward prices (due to the margining systems used by futures exchanges),

but if the asset price is highly negatively correlated with interest rates thenfutures prices tend to be lower than forward prices

Some of the other factors that contribute to there being a differenceinclude:

Transaction costs Taxes Treatment of margins (dont exist for forwards) Liquidity (usually higher for futures than forwards) Default risk (lower for futures)

For most purposes it is reasonable to assume forward and futures prices arethe same, especially for short maturity contracts. For longer maturity

products (e.g. Eurodollar futures) it is less valid to make this assumption.

Empirical studies have been done on this issue:

For currencies these studies found few statistically significantdifferences between futures and forward prices. For commodities (metals) there were statistically significant differences,with futures prices being higher.

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Stock Index Futures

A stock index is an index that measures / reflects the investmentperformance of a portfolio of stocks.

The group of stocks in the portfolio is called the index population. Forinstance the ASX 200 index population is the top 200 companies measured

by market capitalization.

In Australia, the stock exchange publishes a variety of different stockindexes: one for the overall stock market, one for industrial companies,another for the mining and oil industry, and various others for specificindustries. In Australia these indexes are based on weights proportional tomarket capitalization (# shares on issue times market price of the shares) in

the index population. Some overseas indexes (eg Dow Jones) are based onweights proportional to the stock prices for all stocks in the index

population.

There are 2 main types of stock index:a price index and an accumulation index.

Changes in the level of a price index allow us to measure the change in thevalue of the portfolio of stocks. It measures the capital gain type return

only.

An accumulation index is similar except that it is calculated by assumingthat any dividend income paid by the portfolio is re-invested back into the

portfolio of stocks in proportion to the market values of each stock in theindex population.

Changes in the level of an accumulation index measure the total return onthe portfolio from both capital gains and income. Note that this total return

calculation assumes no tax or transaction costs.

Stock index futures are futures contracts where the payoff at maturitydepends on the value of a stock index. These are usually based on a priceindex and not on an accumulation index.

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Futures / Forward Prices for Stock Index Futures:

The dividend income provided by a portfolio of stock is usually spreadmore or less uniformly over a year, as different stocks pay dividends ondifferent dates.

It is normal practice to treat the dividend income as a dividend yieldinstead of as a known cash income. In doing this we should estimate thedividend yield y as the average annualized dividend yield during the life ofthe contract and include those dividends in the calculation for which the exdividend date occurs during the life of the contract.

The formula for the forward price of a stock index for maturity T is( )

0 0

r y TF S e

= where

y is the dividend yield and

0S is the initial value of the stock index.

Contango and Backwardation:

SP500 index futures as at 16/3/2001march 01 117330

june 01 118470

sept 01 119640dec 01 120740march 02 121790

june 02 123040

These futures prices are increasing with maturity, at approx 3.8% p.a. Thisis probably because the risk free interest rate is 3.8% above the dividendyield.

When futures prices are an increasing function of maturity we say themarket is in contango.

When the futures prices are a decreasing function of maturity we say themarket is in backwardation.

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Index Arbitrage

if ( )0 0r y T

F S e> so that the futures price is above the arbitrage free value,

then an arbitrage profit can be made by doing the following:

buy spot the stocks underlying the index short sell the futures contractthis is often done by firms holding short term money market investments

if ( )0 0

r y TF S e

< so that the futures price is below the arbitrage free value,

then an arbitrage profit can be made by doing the following:

short sell the stocks underlying the index take a long position in the futures contractthis is often done by pension funds that hold an index portfolio of stocks

These strategies are often implemented by trading a small representativesample of the stocks in the index instead of the whole index population.This is more efficient and avoids excessive transaction costs. Sometimesthis is done via computerized systems, known as program trading.

Forward and Futures Contracts on currencies

In our notation for assets, in the context of a foreign currency tS means thevalue at time t of 1 unit of the foreign currency as measured in the localcurrency. This is what they call a direct quote.

For instance if the local currency is AUD and the foreign currency is USDthen the exchange rate might be USD 1.00 = AUD 2.00, meaning that the

price of 1 unit of the USD will cost 2.00 Australian dollars. We wouldwrite that as

tS =2.00

For many countries that were formerly part of the British Empire, foreignexchange quotes are indirect quotes against the US dollar instead ofdirect quotes. This means the exchange rate is expressed as the numberof US dollars that 1 unit of the local currency would be worth.

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In the above example, 1 unit of AUD would be worth $0.50 US dollars. In

our notation this would be1

tS

If you are the holder of an amount of foreign currency then you can

deposit that money in a foreign bank account and earn the risk free rate ofinterest that applies in that currency.

Notation:

r means the domestic risk free interest rate fr means the foreign risk free interest rate

Here both rates are taken to be continuously compounded rates and we

assume that both the spot and forward prices are direct quotes.

The relationship between the spot and forward exchange rates is( )

0 0

fr r TF S e

=

this relationship is called covered interest parity

comments on the formula( )

0 0

fr r TF S e

=

if the domestic rate is higher than the foreign rate then the forward pricewill increase with the term to maturity

if the domestic rate is lower than the foreign rate then the forward pricewill decrease with the term to maturity

this formula is really the same as the formula ( )0 0 r y TF S e = if fy r= sowe can view the foreign interest rate as being like a dividend yield

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Example: 2 year forward exchange rate between the USD and AUD

To see where this relationship comes from we shall consider how to createa synthetic forward exchange rate contract for the following example:

the spot exchange rate is 0.62 USD per 1.00 AUD the 2 year risk free interest rate is 5% in Australia and 7% in the USA because of the way the exchange rate is quoted, we shall treat the USD

as the domestic exchange rate and the AUD as the foreign exchangerate, in applying the formula

so we have 0 0.62S = , 0.07r = , 0.05fr = , T = 2 ( ) ( )0.07 0.05 20 0 0.62 0.6453fr r TF S e e = = = this means that we can enter a contract today to exchange 1 unit of the

AUD for 0.6453 units of the USD in 2 years time

To create a synthetic forward exhange rate contract:

Now

borrow 1 unit of AUD at 5% for 2 years immediately exchange this 1 unit of AUD for 0.62 units of USD using

the spot market

invest / lend 0.62 units of the USD in a US bank account or short terminvestment at a rate of 7% for 2 years

net cashflow at time t = 0 is nilIntermediate:

our USD asset grows at the USD risk free rate our AUD liability grows at the AUD risk free rateAt maturity in 2 years:

our AUD borrowing matures for an amount of 0.05 21.00 1.105171e = our USD lending matures for an amount of 0.07 20.62 0.713170e = we pay back our AUD loan (pay 1.105171 AUD) and receive our

USD asset of 0.713170 USD

the exchange rate implicit in this transaction (which happens in 2years) is

0.07 2

0.05 2

0.62 0.7131700.645303

1.00 1.105171

e

e

= =

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This example shows how to create a synthetic forward exchange ratecontract:

borrow 1 unit of the foreign currency (financially equivalent toselling a foreign denominated zero coupon bond) at the foreigninterest rate for a term equal to that of the forward

exchange it at the spot exchange rate for the local currency lend / invest the proceeds (financially equivalent to buying a

domestic zero coupon bond) earning interest at the local interestrate for the same term

wait till the end of the term of the forward contract: the foreignasset and the domestic liability combination is equivalent to anexchange rate transaction at the time when they both mature

commodity futures and forwards

Some commodities (e.g. gold, silver) are held as investment assets as wellas having industrial uses.

There are storage costs (including insurance) associated with holdingthese commodities.

Storage costs can be thought of as a negative income. If U is the pv of allstorage costs during the term of a forward contract then the forward price

for delivery at time T is ( )0 0 .rT

F S U e= +

If the storage cost is proportional to the price of the commodity then wecan think of it as being like a negative dividend yield. If the storage cost asa proportion of the commodity price is u then the forward price for

delivery at time T is( )

0 0

r u TF S e

+=

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consumption commodities

Some commodities (e.g. sugar) are consumption assets and not investmentassets. This means that the motivation for holding these assets is to usethem in some industrial process or to consume them. Holding the asset to

time T is not the same for them as having a forward contract to buy theasset at time T. A confectionary manufacturer cant use sugar futures tomake sweets with but can use physical sugar for this purpose.

For consumption commodities the arbitrage arguments used to deriveformulae for forward prices as in the case of investment assets dont holdexactly.

Suppose that ( )0 0 .rT

K F S U e> = + is the forward price for some

commodity.an arbitrageur can

borrow ( )0S U+ at rate r for term T and buy one unit of the commodity and pay storage costs as they occur and short one futures contractAt time T this will deliver a profit of

N ( ) ( ) ( )0 0value of short futures loan maturity paymentholding payoff

. .rT rT T TS K S S U e K S U e+ + = +

this strategy can be easily implemented for any commodity but the more people do this, the sooner the prices of the commodity and

the forward price will change until this inequality no longer holds and itwont be profitable to adopt this strategy

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Suppose that ( )0 0 .rT

K F S U e< = + is the forward price for some

commodity.

an arbitrageur can

sell the commodity and save the storage costs for( )0S U+ invest / lend the proceeds at the risk free rate for term T go long in one futures contractAt time T this would deliver a profit of

( )0 .rT

S U e K = +

This would work ok for an investment asset. But the problem here is thatselling the commodity and going long a futures contract is not a perfect

substitute for a holding of the commodity if it is a consumption asset.

Hence for commodity futures all we can really say is that ( )0 0 .rT

F S U e +

if storage costs are proportional to the commodity price then ( )0 0r u T

F S e+

Convenience Yields

This is a way of measuring the benefits of holding the physical asset:

If U is the pv of all storage costs during the term of a forward contract thenthe convenience yield

Cy is defined by the equation

( )0 0. .Cy T rT

F e S U e= +

If the storage cost as a proportion of the commodity price is u then theconvenience yield is defined by the equation

( ) ( )0 0 0 0 CCr u y T r u Ty T

F e S e F S e+ +

= =

where 0F is the futures price.

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COST OF CARRY

This means the storage cost plus the interest required to finance purchaseof the asset less any income on the asset. It is another way to think aboutthe relationship between spot and forward prices.

Let c be the cost of carry.

For an investment asset the relationship between spot and forward prices is.

0 0.c T

F S e=

For a consumption asset with a convenience yield we have( )

0 0Cc y TF S e

=

type of asset cost of carry

zero dividend paying stock c r=stock index c r y=

foreign currencyf

c r r=

commodity with storage cost yieldu

c r u= +

Futures price and expected future spot price

What does the futures price say about the expected spot price on thematurity date?

We consider the expected spot price on some future date to be themarkets consensus opinion about the spot price on that future date. Wemean the average of the markets opinion about this matter.

Some people take the view that the best estimate of what the spot pricewill be at time T in the future is the forward or futures price for delivery attime Tthe futures price is the best unbiased estimator of the future spot price

Economists John Maynard Keynes and John Hicks argued that this maynot be true:

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Suppose that hedgers tend to hold short forward positions whilespeculators tend to hold long forward positions. Then the futures price will

be lower than the expected future spot price.

The reasoning behind this conclusion is that

speculators will only go long the forward if they expect to make moneyand require compensation for the risks they are taking on. hedgers on the other hand are willing to lose money on the average

because they see it as the cost of obtaining protection against price falls.

Suppose that hedgers tend to hold long forward positions while speculatorstend to hold short forward positions. Then the futures price will be higherthan the expected future spot price.The reasoning behind this conclusion is that

speculators will only short the forward if they expect to make moneyand require compensation for the risks they are taking on.

hedgers on the other hand are willing to lose money on the averagebecause they see it as the cost of obtaining protection against priceincreases.

speculating on long futures contracts and risk / return

Suppose a speculator goes long a futures contract hoping that it will be

profitable at maturity (i.e. that 0TS F> ). The speculator does the

following:

invests 0. rTF e , the pv of the futures price at the risk free rate for term Tat rate r

goes long in a futures contract waits till maturity receives 0F , the proceeds from the risk free investment buys the stock for amount 0F sells the stock for its market price of TS the profit at maturity from selling the stock is 0TS F the time 0 cashflow from doing this is 0. rTF e the time T cashflow from doing this is ( )0 0T TS F S F = +

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the net present value of this proposed strategy is

( )N

.

0

riskinitialexpected adjustedoutlaypayoff discount

factor

.k T rT TNPV E S e F e =

k here is the discount rate appropriate to the degree of risk in the

investment

In an efficient market the npv from doing this will be zero

( ) ( ) ( ).. 0 00 .r k Tk T rT

T TE S e F e F E S e = =

so the relationship between futures price and the expected future spot

price is ( ) ( ).0r k T

TF E S e

=

The capital asset pricing model (security market line)

( )Mk r E R r = +

where

( )ME R is the expected return on the market portfolio (i.e. thesharemarket).This is always higher than the risk free rateris the risk free rate of interest

M

i

= = the correlation () multiplied by the ratio of the standard

deviation of the return on the stock market and the standard deviation ofthe return on the asset

This equation says that the rate of return on an asset depends on thecorrelation between returns on that asset and the returns on the market

portfolio (which in practice means the stock market) via the beta factor.

If the correlation is zero then ( )0 Tk r F E S = =

If the correlation is > zero then ( )0 Tk r F E S > < If the correlation is < zero then ( )0 Tk r F E S < >

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Futures Contracts and Hedging

There are various differences between futures and forward contracts, aswell as many features in common.

Futures contracts are exchange traded whereas forward contracts are overthe counter contracts.

There is a part of the exchange called the exchange clearing house, whichperforms the functions of matching buyers and sellers and making andreceiving payments to / from market participants. It acts as thecounterparty to every futures market transaction. A holder of a longfutures and a holder of short futures each have a contract with the clearing

house and not with each other.

The exchange clearing house administers a system of deposits andmargins for futures contracts. This means that for futures contracts theremay be various cashflows paid / received prior to the maturity date.Forward contracts dont have this problem.

The system of margins and deposits provides protection to users of themarket against the risk of default by the counterparty. So the level of

counterparty credit risk is lower with futures contracts than with forwardcontracts.

The futures contracts are standardized in their features. This enhances theliquidity of the market for them. Forward contracts are not standardized.

Futures contracts can be reversed easily by entering into an oppositetransaction in the same contract. This is a process known as closing outa futures contract. Forward contracts are much more difficult to reverse

once entered into.

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Sydney Futures Exchange (SFE) Bank Bill Futures Contract

The details of the contract which are standardized include:The quality and quantity of the underlying asset, The maturity datesThe settlement method, The method of quotation of the futures price,

The minimum price movement

Contract Unit:

(underlying

asset)

A$1,000,000 face value 90-Day Bank Accepted Billsof exchange

Contract Months:

(available terms to

maturity)

March/June/September/December up to twentyquarter months or five years ahead

Minimum Price

Movement:

One hundred minus annual percentage yield quoted to

two decimal places.

Last Trading Day:12.00 noon on the business day immediately prior tosettlement day.2

Settlement Day: The second Friday of the delivery month.

Trading Hours: 5.10pm 7.00am and 8.30am 4.30pm2 (during USdaylight saving time)3 5.10pm 7.30am and 8.30am 4.30pm2 (during US non daylight saving time)3

Settlement

Method:

Ten bank accepted bills or ten bank negotiable

certificates of deposit (NCDs) each of face valueA$100,000, or two bank accepted bills or or banknegotiable certificates of deposit each of face valueA$500,000 or one bank accepted bill or EBA or banknegotiable certificate of deposit of face valueA$1,000,000 maturing 85-95 days from settlementday.

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The method for quoting the futures price on the ASX for interest rate

futures.

The quoted futures price is not the same as the cash delivery price (theactual dollar amount you have to pay on the delivery date).

Instead it is quoted as 100Q y= where y is the annual yield to maturity

expressed as a percentage.

For example if the annual yield to maturity is 6%y = then the futures

quote is 100 100 6 94Q y= = = .

To compute the actual cash delivery price we need to compute the yield yfrom the quote Q and then use the yield as the input to the bank billvaluation formula, which is

1000000 1000000985421.1663

90 90 61 1

365 100 365 100

valuey

= = =

+ +

So when the contract matures, you would pay $985,421.17 in cash andreceive a 90 day bank accepted bill with a face value of $1,000,000 and aterm of 90 days. The futures quote was 94 but the cash delivery price is not

94.

The settlement method:

For this contract it is settled by physical delivery (in other words theholder of a long position actually receives a 90 day bill) instead of cashsettlement.

Some contracts are settled by physical delivery but some are instead

settled in cash.

Suppose that you entered a bank bill futures contract today (15 August2005), to buy at a futures quote of 94, for delivery in December 2005.Then the delivery date would be the second Friday of December 2005 (the9

thof December 2005).

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On that date you would be obligated to buy a bank bill with a face value of$1000000 and a maturity date 90 days after 9/12/2005, which happens to

be 9/3/2006). The price of this bill would be $985421.17.

It is possible that on the date when the futures matures, the bank bill yield

has dropped to 5% p.a., so that the bank bill price has risen to1000000

$986,619.8190 5

1365 100

price = =

+

Under this scenario, you can buy the bill for $985421.17 and thenimmediately sell it for $986,619.81.

This makes you a profit of $986,619.81-$985421.17 =1198.65

If the contract were cash settled instead of settled by physical delivery,then on the delivery date the futures exchange would pay you as the holderof the long position in the futures contract, an amount of cash equal to the

profit you could get by immediately selling the asset. Instead of paying$985421.17 and receiving a 90 day bank bill with $1m face value, youwould instead receive $1,198.65 in cash.

For our purposes we shall treat futures and forward contracts as

almost identical.

Example of Using Bill Futures for hedging:

Hedging is like insurance: the motivation is to reduce financial risk. Let ussuppose you are a corporation and you know that you will have to borrow$100m for 90 days in December 2005, to finance the purchase of importedluxury cars for your car dealing business. It is currently 15 August 2005.You dont know what the interest rates on 90 day bills will be in

December. You want to eliminate the risk of an increase in interest rates.Today, 90 day bank bill yields are 6.0%. and the December bill fufurescontract is trading at a quoted futures price of 94.

To protect the firm from an increase in interest rates, you need to enter ashort futures contract to sell bank bills in December. Remember that

borrowing money is financially equivalent to selling a debt security: alender buys and a borrower sells a debt security.

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You can enter a contract today to borrow money for 90 days starting from9/12/2005, at a yield of 6% buy shorting futures contracts. The delivery

price of the $1m face value 90 day bill will be $985,421.17. This is theamount you as the holder of 1 futures contract would receive on the

maturity date of the futures contract which is 9/12/2005. Then when thebank bill matures, 90 days later on 9/3/2006 you would have to pay back$1,000,000.

The amount of money you wanted to borrow in December was $100m, not$985,421.17. To be able to guarantee you could borrow this amount inDecember you would need to short sell n bill futures contracts where

100,000,000.00101.4795 101.5

985,421.17n = =

But you cant buy a fraction of a contract, only a whole number ofcontracts. This is one of the problems with futures contracts: thestandardization of the contract means that the contract may not perfectlymatch the needs of the users.

This is not such a big problem in this example, but suppose you wanted toborrow $0.5m in December instead of $100m. Then the bill futurescontracts standardized amount would be more of a problem.

The payoff from hedging:

Let us ignore the complication that we cant trade 101.4795 contracts andpretend that we can trade the number of contracts that we want. InDecember when the futures matures, it is possible that 90 day bank billinterest rates have risen to 7%. This would mean that for $1m face value,the sale proceeds would be $983,032.59 instead of $985,421.17.

Buy selling bank bill futures, we have been able to save the firm an

amount of ( )$242,391.60 =101.4795 $985,421.17 - $983,032.59 Compared with the amount of money we could have raised by selling101.4795m of 90 day bills in the physical market

( ) ( )Tpayoff = number of futures contracts X - S

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X = delivery price of futures contract TS = price of the underlying asset (a 90 day bank bill) at the maturity

date of the futures

Deposits and margins

The futures exchange requires that anyone who wants to trade a futurescontract (either long or short) must open a special type of bank accountwith the exchange clearing house. This is called the margin account.

The first deposit is required on the day when the futures contract isinitiated. It is called the initial deposit or the initial margin. It is anamount of money equal to the maximum likely price fluctuation in thecontract over a 1 day period. The exchange decides the amount of the

initial deposit. It is usually approximately 5% of the value of theunderlying asset.

At the end of each day, every futures contract is revalued according to theclosing futures price. This process is called marking to market.

It is possible that your futures position may have increased in value or itmay have decreased in value. If it has decreased in value, the exchangewill require you to pay an additional amount into the margin account. Thisamount is called the variation margin also known as a maintenancemargin. It is usually equal to the amount by which your contract hasfallen in value over the last 1 day.

Example: long bank bill futures position

For the bank bill futures contract above, let us assume that1.the initial deposit is 5% of the value of the underlying asset and2.

that we are long the futures (not short as above).

The initial futures quote was 94.00 and the implied yield to maturity was6%, so the delivery price is $985,421.17

The initial deposit would be 985421.17 0.05= $49,271.06=

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Let us suppose that on day 2, the futures quote for this contract changesfrom 94.00 to 93.80. This means the yield to maturity has changed to6.20% and the futures price has changed to $984,942.52. The value of ourlong futures contract has fallen by an amount $985,421.17 - $984,942.52= $478.64

We are required to pay in a variation margin of $478.64, taking theamount in the margin account to $49,749.70

Now suppose that on day 3, the futures quote changes to 93.70. Thismeans a yield of 6.30% and the delivery price changes to $984,703.37.The value of our long futures position has fallen again, this time by anamount of $239.15.The variation margin we would have to pay is $239.15

Example: closing out the contract

On day 4, the futures quote falls again to 93.50. You went long theDecember bill futures contract because you thought that bill yields weregoing to fall but instead they have been rising, the prices of bills fordelivery in December has been falling. You decide you want to cut yourlosses and get out of the contract.

To do this you have to enter into a contract over the same underlying asset(the 90 day bill) with the same maturity (December 2005) but opposite indirection (i.e. short instead of long).

To cancel out your long December bill futures at 94.00 you have to goshort another December bill futures at a quote of 93.50.

Under the long futures contract you are obligated to buy a 90 day bank billon 9 december 2005 for a price of $985,421.17.

Under the short futures contract you are obligated to sell a 90 day bank bill

on 9 december 2005 for a price of1000000

$984,225.4390 6.5

1365 100

=

+

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LetT

Sbe the actual price of a 90 day bill on 9 december 2005. Then the

payoff from this combination of long and short futures contracts is

( ) ( )T Tlong futures payoff short futures payoff

S - $985,421.17 $984,225.43-Spayoff = +

note that this is independent of TS $984,225.43-$985,421.17= -$1,195.74payoff =

This is a loss of $1195.74 and it is independent of the actual price of a billon the maturity date. Your future obligations on the 2 contracts cancel outto this net amount.

Your margin account would have an amount of $49,988.85 so the losswould be taken out of your margin account. The remaining balance of the

margin account would be returned to you.

This process of matching off 2 opposite transactions in the same asset forthe same maturity date is called closing out the contract.

By having this system of deposits and margins, the exchange clearinghouse protects itself (and hence protects futures market traders) fromdefault by a party to futures contract. If a traders position is makinglosses, there is the possibility that they will default. If they cant pay themargin call (the variation margin) then the exchange will close out theirfutures contract as above and take the cost of doing so out of their marginaccount.

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Forward Rate Agreement (FRA)

This is an over the counter contract that a certain fixed interest rate willapply to a certain fixed principal amount during a specific future period oftime.

Suppose that

the principal amount is L the FRA contract specifies that the holder (a financial institution say)

will receive interest at rateK

R on the amount L, for the period from

time 1T to time 2T

FR = the forward LIBOR interest rate for the period from time 1T to

time 2T

R = the actual LIBOR interest rate observed at time 1T for a depositmaturing at time 2T , and which has a term of 2 1T T

The cashflows to the holder of the FRA are:

L at time 1T (cash outflow from lending / investing) and ( )2 11 KL R T T+ + (cash inflow from maturity proceeds) at time 2T

Valuing the FRA

At zero cost it is possible at time 0 to enter into an arrangement toborrow L at time 1T and repay it at time 2T with interest at the forward

interest rateFR (the forward rate, as observed at time 0 for a loan over

the period from time 1T to time 2T )

by combining this forward borrowing with the above FRA we cangenerate a situation with the following cashflows:

1Time T 0cashflow L L= + =

( ) ( )( )( )

2 2 1 2 1

2 1

Time T 1 1K F

K F

cashflow L R T T L R T T

L R R T T

= + + +

=

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The value of the FRA is the present value of this cashflow at time 2T

The value of the FRA is thus

( )( ) ( )2 1 2 2.exp .value K F FRA L R R T T R T = where ( )2 2 2(0, ) exp .P T R T = is the price at time 0 of a zcb maturing attime 2T (same as the discount factor at the risk free interest rate)

More about hedging with futures:

A hedge is a transaction (or set of transactions) which is intended to offsetthe risk of loss due to changes in interest rates, exchange rates, financialmarket prices or commodity prices. If it works, it works by generating asituation where changes in the value of the hedge instrument are in the

opposite direction to changes in the variable being hedged.

Perfect Hedge:

A perfect hedge is one that completely eliminates risk. It is rare to find aperfect hedge. An example of one is closing out the long bank billfutures contract by entering into a short futures contract, as describedabove.

Buy and Hold (or set and forget)

Sometimes a financial instrument exists that hedges the exposure and afteryou enter into the transaction you can hold your position until maturitywithout the need for any adjustments during the period. This is a buy andhold strategy.

Short Hedge:

You own some asset (e.g. gold) and you expect to sell it at some futuretime. You are worried about the possibility of the price falling betweennow and when you expect to sell it. You can hedge against this risk byshorting futures contracts on gold now.

If the price of gold does fall, you make a gain on the futures that offsetsthe loss on the holding of physical gold.

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You are a gold miner and you expect to have 100000oz of gold ready tosell in 12 months time. The current gold price is $400 / oz.

If the gold price were to fall to $300 in a years time you will make a loss

of $100 / oz compared with todays prices.

You could hedge against this risk by entering into a futures contract to sellgold at say $420 / oz in 12 months time. If the price of gold did fall to$300 then the futures contract would be in profit by$120 = $420-$300. This profit offsets the loss of $100 on the physicalgold.

Long Hedge

This is appropriate if you know you will have to buy some commodity atsome future time and you are concerned about the price of that commodityrising too much. For example plastics manufacturer needs to buy oil tostay in business. If it has fixed price contracts to supply customers it isexposed to loss if the price of oil goes up. It can reduce the extent of thisloss by going long in oil futures contracts.

Motivations for hedging and arguments for and against it.

Most corporations are not in the business of forecasting or trading on thebasis of changes in interest rates, exchange rates, financial market pricesand commodity prices. They are in the business of producing goods andservices. Their profits may be adversely impacted by changes in the abovevariables.

It makes sense for them to hedge the risks associated with these variablesas they arise. They can then focus on their core business activities, where

they do have the requisite expertise. Hedging will help to avoid nastysurprises such as an increase in the cost of their inputs.

In practice many risks are left unhedged. There are various reasons forthis.

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Shareholders and risk:

In some cases the shareholders may be able to hedge the risk themselves,independently of the company and they dont need the company to do itfor them. For instance, a gold miner / manufacturer could hedge most of

their gold price risk. Doing this would transform the returns from thebusiness into an almost risk free operation. But the shareholders whobought the shares may have done so precisely because they wanted tospeculate on gold prices. They might not want all risk to be hedged away.The company may be punished by a falling share price if the shareholdersfeel this way.

There is a cost associated with hedging:

There may be explicit costs associated with hedging: up front costs ofpurchasing financial instruments, brokerage, commission, margin calls onfutures etc. There is also opportunity cost. If you hedge with futures thenyou are protected against upside risk (profit) as well as downside risk(loss).

Industry practice:

In some industries it is normal for prices to fluctuate up and down in

response to changes in the costs of the raw materials and other inputs. Thissituation is a natural hedge for the firms profit because changes in therevenues and changes in the expenses offset each other. In this situation, afirm which hedges the costs of these raw materials will actually be worseoff than one that does not.

Hedging may increase risk instead of reducing it.

Basis Risk:

In practice, users of financial derivatives often dont know the precise datein the future when an asset may have to be bought or sold. Even if they do,it may not coincide with the maturity date for the available futurescontracts.

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The complications involved in hedging include

the asset price to be hedged is not exactly the same as the assetunderlying the futures contract (e.g. a 2 year bond is not matched by the

bank bill contract or the 3 year bond futures contract)

the quantity of the asset to be hedged is not an exact integer multiple ofthe asset underlying the futures contract the exact date when the asset is to be bought or sold is not known with

certainty or it does not match the maturity date of available futures

the hedge may require that the futures contract be closed out before itsexpiration date

These complications give rise to what is known as basis risk

The basis is defined as

basis = spot price of assetbeing hedged

- futures price ofcontract being used

If the asset being hedged and the underlying asset for the futures are thesame (in both quality and quantity) then the basis will be zero at theexpiration date of the futures contract.

Note that the spot price of the underlying asset and the futures price will bethe same at the expiration date of the futures contract. For a proof of thissee section 2.3 page 23-24 of Hulls book (5

thedition).

Before expiration the basis may be either positive or negative.

When the spot price increases by more than the increase in the futures

price the basis increases referred to as strengthening the basis.

When the spot price increases by less than the increase in the futures price,the basis decreases referred to a weakening the basis.

Notation:

1 1spot price at time tS =

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2 2spot price at time tS =

1 1futures price at time t for a futures that matures at time TF =

2 2futures price at time t for a futures that matures at time TF =

1 2t

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Suppose a hedger wants to sell the asset at time t2 and takes out a shortfutures contract at time t1.

The time t2 cashflow is ( ) ( )* *2 1 2 1 2 2 2 2S F F F S F S S + = + + in this case the basis is made up of 2 components

( ) ( )* *2 2 2 2 2basis if assets differencewere the same between

2assets

b S F S S = +

Choice of contract

A key decision is what futures contract to use for hedging. Need to

consider what maturity date for the futures? which futures contract has the best underlying asset?If there is a futures contract with an asset that is the same as the asset beinghedged then the choice is easier.

If this is not the case then we need to consider which of the availablefutures contracts has futures prices that have the highest correlation with

the price of the asset to be hedged.

The normal choice for the maturity date is to pick the maturity date closesto but later than the date when the asset is to be bought / sold.

Basis risk tends to increase with the time difference between the futuresmaturity date and expiration date of the hedge.

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Minimum Variance Hedge Ratio

The hedge ratio is the ratio of the size of the position taken in futurescontracts to the size of the exposure. It is an important concept inderivative pricing and we will meet it again when we look at options.

Notation

S =change in the spot price S during the term of the hedgeF =change in the futures price F during the term of the hedge

S = standard deviation of S

F = standard deviation of F

= correlation between S and F

the optimal hedge ratio is given by the formula

* S

F

h

=

Proof: suppose the hedger is long the asset and short h units of the futurescontract: then

the change in the value of the hedged position over the life of the hedgeis S h F for each unit of the asset held

the variance of the change in the value is( ) ( ) ( ) ( )var var var 2cov ,S h F S h F S h F = +

( )2 2 2 2 var S F S F h h h F S + = this variance is a function of h, the hedge ratio

( ) 2 2 2var 2S F S F g h h h = = + we can minimize this by choosing the right value of h. The right value

of h is found by differentiating this function of h and equating thederivative to zero

( ) 2 22 2 0 S F SF S FF F

dg h h h

dh

= = = =

note that ( ) ( )2 2 2var var Fh F h F h = =

and that ( ) ( )cov , .cov , . . .S FS h F h S F h = =

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if we have a situation where 1 and 1SF

= = then we have a perfect

hedge and the hedge ratio is 1.00

if we have a situation where 1 and 0.50S

F

= = then hedge ratio =

0.50. The futures price always changes by twice the change in the spotprice, so half a futures contract is a perfect hedge.

The hedge effectiveness is defined as the proportion of the variance that is

eliminated by hedging. This is ( )2

22 * F

S

h

=

Optimal Number of futures contracts:

Notation:

AN = size of position being hedged in units

FQ = size of one futures contract in units

*N = optimal number of futures contracts for hedging

the number of futures contracts for hedging is

* * A

F

NN h

Q=

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Numerical example:

An airline needs to buy 2 million gallons of jet fuel in 2 months anddecides to use heating oil futures for hedging.

Based on historic data we have S = 0.0263 = standard deviation of change in jet fuel price S F = 0.0313 = standard deviation of change in heating oil futures price

F

= 0.0928 correlation between S and F 0.0263* 0.928 0.78

0.0313S

F

h

= = =

the number of units of jet fuel being hedged is 2,000,000AN = gallons the number of units of heating oil underlying the futures is

42,000F

Q = gallons

the optimal number of futures contracts is* * 20000000.78 37.14 37

42000A

F

NN h

Q= = =

Stock Index Futures:

If the index underlying the futures contract is a good proxy for the marketportfolio then the optimal hedge ratio to hedge a portfolio of shares is thesame as the beta factor for that portfolio of shares.

The optimal number of futures contracts to hedge an equity portfolio is

* PN

A= where

P is the current value of the portfolio A is the current value of the stocks underlying one futures contract

This is really the same as the formula * * A

F

NN h

Q= above but written in

different notation.

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Example:

the S&P 500 index value = $1,000 = F1 value of equity portfolio to be hedged = $5,000,000 = S1 risk free interest rate = 10% p.a. (continuously compounded) dividend yield on index = 4% beta of portfolio = 1.50 we want to hedge the portfolio value over the next 3 months by using a

futures that matures in 4 months

1 futures contract is for delivery of $250 times the indexThe futures price should be

( )4

0.10 0.0412

1 1000 1020.20F e

= =

the number of futures contracts to be shorted to hedge the portfolio is

* 5,000,0001.5 30250,000

PN

A= = =

Scenario 1: in 3 months time the index is at 900.

The futures price will then be( )

10.10 0.04

122 900 904.51F e

= =

the gain on the short futures position is

( ) ( )1 230 250 30 1020.20 904.51 250 867,676F F = =

the loss on the index is 10% (it fell from 1000 to 900) the index dividend yield is 4% p.a. (approx 1% per 3 months) the return on the index is therefore approx -9% over 3 months the risk free rate of interest is approx 2.5% over 3 monthsWe can estimate the return on the equity portfolio from the return on theindex by using the capm equation

( )( ) ( )0.025 1.5 0.09 0.025 0.1475Mk r E R r = + = + =

the expected value of the equity portfolio at time 3 months is therefore

( )2 5,000,000 1 0.1475 $4,262,500S = + = The expected hedged equity portfolio value is therefore

( )2 1 2 $4,262,500 $867,676 $5,130,176S F F+ = + =

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Reasons for hedging equity portfolio

If you expect the index to fall in value then it will probably take the valueof your portfolio down with it. An alternative to hedging with futures is tosell the portfolio now and buy it back later. However the transaction costs

would be huge. We can achieve the same thing using futures at a muchlower transaction cost, and it takes less time to put the transaction intoeffect.

Another reason is that you may believe that your portfolio will outperformthe index. By shorting futures and being long the equity portfolio, thehedger is exposed only to the performance of the portfolio relative to themarket index.

Sometimes a portfolio manager may want to change the beta of theportfolio as part of active portfolio management. If they take the view thatthe market is going to go up, then they might want to increase the portfolio

beta. One way to do this is to sell low beta stocks and buy high beta stocks.This involves a lot of trades and the associated transaction costs.Alternatively it can be achieved much more quickly and at lowertransaction cost by using index futures.

To change the portfolio beta from to *

when > * we need to take a short position in ( )*PA

futures

contracts

when < * we need to take a long position in ( )*PA

futures

contracts

hedging exposure to an individual stock

On the SFE there are individual share futures contracts for some (but notall) stocks traded on the ASX. If so we can use this to hedge. If not you

need to hedge by shortingP

index futures where is the beta of that