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Chapter 14: Future Chapter 14: Future PricesPrices

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

Objective•How to price forward and futures

•Storage of commodities•Cost of carry

•Understanding financial futures

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Chapter 14: ContentsChapter 14: Contents

1 Distinction Between Forward & 1 Distinction Between Forward & Futures ContractsFutures Contracts

2 The Economic Function of Futures 2 The Economic Function of Futures MarketsMarkets

3 The Role of Speculators3 The Role of Speculators

4 Relationship Between Commodity 4 Relationship Between Commodity Spot & Futures PricesSpot & Futures Prices

5 Extracting Information from 5 Extracting Information from Commodity Futures PricesCommodity Futures Prices

6 Spot-Futures Price Parity for Gold6 Spot-Futures Price Parity for Gold

7 Financial Futures7 Financial Futures

8 The “Implied” Risk-Free Rate8 The “Implied” Risk-Free Rate

9 The Forward Price is not a 9 The Forward Price is not a Forecast of the Spot PriceForecast of the Spot Price

10 Forward-Spot Parity with Cash 10 Forward-Spot Parity with Cash PayoutsPayouts

11 “Implied” Dividends11 “Implied” Dividends

12 The Foreign Exchange Parity 12 The Foreign Exchange Parity RelationRelation

13 The Role of Expectations in 13 The Role of Expectations in Determining Exchange RatesDetermining Exchange Rates

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14.1 Distinction Between 14.1 Distinction Between Forward & Futures Forward & Futures ContractsContracts• parties agree to exchange some item in the future at a parties agree to exchange some item in the future at a

delivery price specified nowdelivery price specified now

• the forward price is defined as the delivery price which the forward price is defined as the delivery price which makes the current market value of the contract zeromakes the current market value of the contract zero

• no money is paid in the present by either party to the no money is paid in the present by either party to the otherother

– the face value of the contract is the quantity of the item the face value of the contract is the quantity of the item specified in the contract multiplied by the forward pricespecified in the contract multiplied by the forward price

– the party who agrees to buy the specified takes the long the party who agrees to buy the specified takes the long position, and the party who agrees to sell the item takes the position, and the party who agrees to sell the item takes the short positionshort position

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TermsTerms

– Open, High, Low, Settle, Change, Open, High, Low, Settle, Change, Lifetime high, Lifetime low, Open Lifetime high, Lifetime low, Open interestinterest

– Mark-to-marketMark-to-market

– Margin requirementMargin requirement

– Margin callMargin call

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Characteristics of FuturesCharacteristics of Futures

• Futures are:Futures are:– standard contractsstandard contracts

– immune from the credit worthiness of immune from the credit worthiness of buyer and seller becausebuyer and seller because• exchange stands between tradersexchange stands between traders

• contracts marked to market dailycontracts marked to market daily

• margin requirementsmargin requirements

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14.2 The Economic 14.2 The Economic Function of Futures Function of Futures MarketsMarkets

• The futures markets facilitate the The futures markets facilitate the re-allocation of exposure to re-allocation of exposure to commodity price risk among market commodity price risk among market participants participants

• But:But:

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– by providing a means to hedge the by providing a means to hedge the price risk associated with storing a price risk associated with storing a commodity, futures contracts make it commodity, futures contracts make it possible to separate the decision of possible to separate the decision of whether to physically store a whether to physically store a commodity from the decision to have commodity from the decision to have financial exposure to price changesfinancial exposure to price changes

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The Economic Function of The Economic Function of Futures Markets Futures Markets (Continued)(Continued)

• A distributor, j, may hedge byA distributor, j, may hedge by– selling the commodity on the spot market selling the commodity on the spot market

now at a price Snow at a price S

– selling short a futures contract at a price F selling short a futures contract at a price F and deliver the commodity at a specified and deliver the commodity at a specified time in the futuretime in the future• there will be a carrying cost Cthere will be a carrying cost Cj j for for

distributor j, and she will store only if Cdistributor j, and she will store only if Cjj < F < F - S- S

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The Economic Function of The Economic Function of Futures Markets Futures Markets (Continued)(Continued)

• The difference between the futures The difference between the futures price and the spot price, F - S, is price and the spot price, F - S, is called the called the spreadspread, and governs how , and governs how much wheat will be stored, and by much wheat will be stored, and by whomwhom

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The Economic Function of The Economic Function of Futures Markets Futures Markets (Continued)(Continued)

• Suppose the commodity is wheat, Suppose the commodity is wheat, and next year’s crop is expected to and next year’s crop is expected to be much higher than average, then be much higher than average, then futures prices may be lower than futures prices may be lower than the spot, (the spread may be the spot, (the spread may be negative,) nobody will store wheatnegative,) nobody will store wheat

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The Economic Function of The Economic Function of Futures Markets Futures Markets (Continued)(Continued)

• The existence of the futures market The existence of the futures market for wheat conveys information to for wheat conveys information to allall producers, distributors, and producers, distributors, and consumers; and this eliminates the consumers; and this eliminates the necessity for market participants to necessity for market participants to gather and process information in gather and process information in order to forecast the future spot order to forecast the future spot priceprice

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14.3 The Role of 14.3 The Role of SpeculatorsSpeculators

– HedgerHedger• anyone using a futures market to reduce anyone using a futures market to reduce

riskrisk

– SpeculatorSpeculator

• anyone who takes a position in the market anyone who takes a position in the market (increasing his risk) in order to profit from (increasing his risk) in order to profit from his forecasts of future spot priceshis forecasts of future spot prices

– (A producer, distributor or consumer who (A producer, distributor or consumer who chooses not to hedgechooses not to hedge her risk may be her risk may be considered to be a speculator)considered to be a speculator)

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The Role of Speculators: The Role of Speculators: ExampleExample

– Suppose that the current 1-month futures in Suppose that the current 1-month futures in wheat is $1.5/bushel, and a farming family wheat is $1.5/bushel, and a farming family with stored wheat believes that the price will with stored wheat believes that the price will rise to $2.00rise to $2.00

– NotNot hedging the stored wheat results in the hedging the stored wheat results in the family being exposed to the vagrancies of family being exposed to the vagrancies of the wheat market, and it becomes, in effect, the wheat market, and it becomes, in effect, a wheat speculator (just like their cobbler a wheat speculator (just like their cobbler cousins who are long wheat futures)cousins who are long wheat futures)

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The Role of Speculators: The Role of Speculators: Gamblers and WastersGamblers and Wasters

• Critic: “Speculators have no social value”Critic: “Speculators have no social value”

• Answer:Answer:– Successful speculators make the market Successful speculators make the market

• more efficient as an information resourcemore efficient as an information resource

• provide liquidity when it is needed, which is when provide liquidity when it is needed, which is when producers, distributors, and consumers can’t or won’t producers, distributors, and consumers can’t or won’t hedgehedge

• more efficient by contributing towards recovering the more efficient by contributing towards recovering the fixed costs of providing a futures exchangefixed costs of providing a futures exchange

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14.4 Relationship 14.4 Relationship Between Commodity Spot Between Commodity Spot and Futures Pricesand Futures Prices

• Arbitrageurs place an upper bound Arbitrageurs place an upper bound on futures prices by locking in a on futures prices by locking in a sure profit on futures prices if the sure profit on futures prices if the spread between the futures price spread between the futures price and spot price becomes greater and spot price becomes greater than the cost of carry, F - S than the cost of carry, F - S C C– the cost of carry varies as a function of the cost of carry varies as a function of

time and warehousing organizationtime and warehousing organization

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14.5 Extracting 14.5 Extracting Information from Information from Commodity Futures PricesCommodity Futures Prices

• Case 1 If (Futures Price < Current Case 1 If (Futures Price < Current Spot)Spot)– Then the futures price is an indicator of Then the futures price is an indicator of

the expected future spot pricethe expected future spot price• The futures price is a biased estimate The futures price is a biased estimate

because there are risk premiums and because there are risk premiums and discounts associated with holding the discounts associated with holding the commoditycommodity

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Extracting Information Extracting Information from Commodity Futures from Commodity Futures PricesPrices

• Case 2 If (Futures Price > Current Case 2 If (Futures Price > Current Spot)Spot)– Then the futures price is Then the futures price is notnot an an

indicator of the expected future spot indicator of the expected future spot priceprice• The spread cannot exceed the cost of The spread cannot exceed the cost of

carry carry

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14.6 Spot-Futures Price 14.6 Spot-Futures Price Parity for GoldParity for Gold

• In the case of gold futures, arbitrage In the case of gold futures, arbitrage establishes an upper- and lower-establishes an upper- and lower-bound on the spread between the bound on the spread between the futures and spot prices, resulting in futures and spot prices, resulting in the the spot-futures price-parity spot-futures price-parity relationshiprelationship

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Spot-Futures Price Parity Spot-Futures Price Parity for Goldfor Gold

• There are two ways to invest in goldThere are two ways to invest in gold• buy an ounce of gold at Sbuy an ounce of gold at S00, store it for a , store it for a

year at a storage cost of $h/$Syear at a storage cost of $h/$S00, and , and sell it for Ssell it for S1 1

• invest Sinvest S00 in a 1-year T-bill with return r in a 1-year T-bill with return rff, , and purchase a 1-ounce of gold forward, and purchase a 1-ounce of gold forward, F, for delivery in 1-yearF, for delivery in 1-year

00

1)(

0

01 1 ShrFrS

FSrrh

S

SSffsynAuAu

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Spot-Futures Price Parity Spot-Futures Price Parity for Goldfor Gold

• A contract with life T:A contract with life T:

• This is not a causal relationship, but This is not a causal relationship, but the forward and current spot jointly the forward and current spot jointly determine the marketdetermine the market

• If we know one, then the rule of one If we know one, then the rule of one market determines that we know the market determines that we know the otherother

01 ShrF Tf

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Spot-Futures Price Parity Spot-Futures Price Parity for Goldfor Gold• The following diagram shows how to The following diagram shows how to

create synthetic gold, T-bills, or gold create synthetic gold, T-bills, or gold forward contract from the other twoforward contract from the other two

• All prices are predetermined, All prices are predetermined, – except the price of the one year of the except the price of the one year of the

forward and the price in one year of forward and the price in one year of the gold, but the difference between the gold, but the difference between them is equal to the known financing them is equal to the known financing and storage costsand storage costs

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Rule of One Price: No Rule of One Price: No Arbitrage ProfitsArbitrage Profits

Purchase Actual Au

Sell T-Bill

Sell Au Forward

Sell Actual Au

Settle T-Bill

Settle Au Forward

•Au = Gold•Au = Gold

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Implied Cost of CarryImplied Cost of Carry

• As a consequence of the forward-spot As a consequence of the forward-spot price parity relationship, you can’t price parity relationship, you can’t extract information about the extract information about the expected future spot price of gold expected future spot price of gold (unlike one wheat case) from futures (unlike one wheat case) from futures pricesprices

• The implied cost of carry (per $spot) The implied cost of carry (per $spot) is is h = (F - Sh = (F - S00)/S)/S00 - r - rff

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14.7 Financial Futures14.7 Financial Futures

• We now focus on financial futuresWe now focus on financial futures– standardized contracts for future delivery standardized contracts for future delivery

of stocks, bonds, indices, and foreign of stocks, bonds, indices, and foreign currency currency

– they have no intrinsic value, but represent they have no intrinsic value, but represent claims on future cash flowsclaims on future cash flows

– they have very low storage coststhey have very low storage costs

– settlement is usually in cashsettlement is usually in cash

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Financial FuturesFinancial Futures

• With no storage cost, the relationship With no storage cost, the relationship between the forward and the spot isbetween the forward and the spot is

• Any deviation from this will result in Any deviation from this will result in an arbitrage opportunityan arbitrage opportunity

TfrF

S

1

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Financial Futures: Financial Futures: ExampleExample

• Consider shares in Bablonics, Inc, Consider shares in Bablonics, Inc, trading at $50 each, ($5,000 for a trading at $50 each, ($5,000 for a round lot); assume 6-month T-bills round lot); assume 6-month T-bills yield 6% (compounded yield 6% (compounded semiannually)semiannually)

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Bablonics, Inc (Continued)Bablonics, Inc (Continued)

• 1 Purchase one round lot of stock at 1 Purchase one round lot of stock at spotspot– This results in a negative cash flow This results in a negative cash flow

today of $5,000 (out), and will today of $5,000 (out), and will generate a cash flow of 100*Spotgenerate a cash flow of 100*Spot6m 6m (in)(in)

in six monthsin six months

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Bablonics, Inc (Continued)Bablonics, Inc (Continued)

• 2 Cover today’s negative cash flow 2 Cover today’s negative cash flow by selling short $5,000 worth of 6-by selling short $5,000 worth of 6-month T-bills with a face value of month T-bills with a face value of 5000 (1+ 0.06/2)^0.5 = $5,1505000 (1+ 0.06/2)^0.5 = $5,150

• The cash flow today is $5,000 (in), The cash flow today is $5,000 (in), and the cash flow in six months and the cash flow in six months time will be $5,150 (out)time will be $5,150 (out)

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Bablonics, Inc (Continued)Bablonics, Inc (Continued)

• 3 Cover the risk exposure by selling 3 Cover the risk exposure by selling 100 shares forward at the 100 shares forward at the equilibrium price of equilibrium price of 5000*(1+0.06/2)^0.5 = $5,1505000*(1+0.06/2)^0.5 = $5,150– There is no cash flow today, but the There is no cash flow today, but the

value of this forward contract in six value of this forward contract in six months time will be $(Spotmonths time will be $(Spot6m6m - 5,150) - 5,150)

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Bablonics, Inc (Continued)Bablonics, Inc (Continued)

• -$5,000 (long stock) + $5,000 -$5,000 (long stock) + $5,000 (short bond) + $0 (short forward) = (short bond) + $0 (short forward) = $0$0

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Bablonics, Inc (Continued)Bablonics, Inc (Continued)

• Cash Flow in 6-Months Cash Flow in 6-Months

+ $Spot+ $Spot6m6m (settle long stock) - $5,150 (settle long stock) - $5,150 (settle short bond) +($5,150 - $Spot(settle short bond) +($5,150 - $Spot6m6m) ) (settle forward) = $0(settle forward) = $0

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Bablonics, Inc Bablonics, Inc ((Conclusion)Conclusion)• If your net risk-free investment was If your net risk-free investment was

zero, zero, – and you receive nothingand you receive nothing

• that is what you should expectthat is what you should expect

– and you expect to: and you expect to: • received positive value with no risk, then received positive value with no risk, then

the rule of one price has been violatedthe rule of one price has been violated

• lose value with no risk, then reverse the lose value with no risk, then reverse the direction of all transactions, and again you direction of all transactions, and again you profit with no riskprofit with no risk

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14.8 The “Implied” Risk-14.8 The “Implied” Risk-Free RateFree Rate

• Rearranging the formula, the implied Rearranging the formula, the implied interest rate on a forward given the spot interest rate on a forward given the spot isis

• This is reminiscent of the formula for the This is reminiscent of the formula for the interest rate on a discount bondinterest rate on a discount bond

0

0

1

0

1, T if;1S

SFr

S

Fr

T

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14.9 The Forward Price is 14.9 The Forward Price is not a Forecast of the Spot not a Forecast of the Spot PricePrice• Following the diagrams in Chapter 12 we Following the diagrams in Chapter 12 we

might suppose that the expected price of might suppose that the expected price of a stock isa stock is

• If this is indeed correct, then the forward If this is indeed correct, then the forward price is not an indicator of the expected price is not an indicator of the expected spot price at the maturity of the forwardspot price at the maturity of the forward

FeSeS trtr

sf

Sf

t

02

0

2

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The Forward Price is not a The Forward Price is not a Forecast of the Spot PriceForecast of the Spot Price

• The forward price is obtained The forward price is obtained without risk from the current spot without risk from the current spot and riskless bondand riskless bond

• The spot value at a future date is The spot value at a future date is obtained by investing in the security obtained by investing in the security and accepting (market) risk, and and accepting (market) risk, and this risk must be rewarded this risk must be rewarded

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14.10 Forward-Spot Parity 14.10 Forward-Spot Parity with Cash Payoutswith Cash Payouts

• So far we have assumed that there is no So far we have assumed that there is no dividenddividend– Now suppose that everybody expects an Now suppose that everybody expects an

uncertain dividend in 1 year of Duncertain dividend in 1 year of D

– It is not possible to replicate D because of It is not possible to replicate D because of this uncertaintythis uncertainty

– We will treat D as if it were known with We will treat D as if it were known with certainty, and only deal with 1-year forwardscertainty, and only deal with 1-year forwards

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Forward-Spot Parity with Forward-Spot Parity with Cash PayoutsCash Payouts

• The SThe S00 - F relationship becomes - F relationship becomes

• Note: (forward price > the spot price) Note: (forward price > the spot price) if (D < r S)if (D < r S)

• Because D is not known with certainty, Because D is not known with certainty, this is a this is a quasi-arbitragequasi-arbitrage situation situation

DrSSFr

FDS

10

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14.11 “Implied” Dividends14.11 “Implied” Dividends

• From the last slide, we may obtain From the last slide, we may obtain the implied dividendthe implied dividend

FSrD 1

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14.12 The Foreign 14.12 The Foreign Exchange Parity RelationExchange Parity Relation

• Recall from Chapter 2 the following Recall from Chapter 2 the following diagram:diagram:

Exchange Rate Example

15000 ¥(Borrowed)

15450 ¥ 15450 ¥(Repaid)

£100(Invested)

£109(Matures)

Time

3% ¥/¥ (direct)

3% ¥/£/£/¥

•150 ¥/£

9%£/£

Forward ¥/£

Japan U.K.

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The Foreign Exchange The Foreign Exchange Parity RelationParity Relation

• We used the diagram to show that We used the diagram to show that

• Recall there is a time structure of Recall there is a time structure of interest, and the appropriate risk interest, and the appropriate risk free rate should be usedfree rate should be used

ttY$ r1

Yenfor Spot dDenominate $

r1

Yen on Forward ddenominate $

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14.13 The Role of 14.13 The Role of Expectations in Expectations in Determining Exchange Determining Exchange RatesRates

– Consider a world in which there are two Consider a world in which there are two countries, Domestic & Foreign, and countries, Domestic & Foreign, and conditions are such in each country that conditions are such in each country that the the yield curves are flat, with yields of the the yield curves are flat, with yields of 5% and 10% respectively5% and 10% respectively

– Further assume that the exchange rate is Further assume that the exchange rate is 1 today1 today

– The 1-year forward is The 1-year forward is 1*1.05/1.10=0.95451*1.05/1.10=0.9545

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The Role of Expectations The Role of Expectations in Determining Exchange in Determining Exchange RatesRates

– If the interest rate in Foreign is higher than If the interest rate in Foreign is higher than in Domestic, one explanation may be that in Domestic, one explanation may be that the rate of inflation is higher.the rate of inflation is higher.

– Assume no taxes, and the interest rate Assume no taxes, and the interest rate difference is the result inflation being 5% difference is the result inflation being 5% and 10% respectivelyand 10% respectively

– Then the price dynamics of both countries Then the price dynamics of both countries will result in an exchange rate of 0.9545 will result in an exchange rate of 0.9545 next year, which is also the forward ratenext year, which is also the forward rate

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The Role of Expectations The Role of Expectations in Determining Exchange in Determining Exchange RatesRates

– In real life, things are not so simple, but In real life, things are not so simple, but several mechanisms may be postulated several mechanisms may be postulated that support the that support the expectations expectations hypothesishypothesis

– International investor confidence, and International investor confidence, and their forecasts of inflation, place price their forecasts of inflation, place price pressure on both spot and forward pressure on both spot and forward exchange rates through the exchange rates through the international bond marketinternational bond market