Topic 4 Futures Pricing II

8/9/2019 Topic 4 Futures Pricing II

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Topic 4Topic 4

Futures and Forwards IIFutures and Forwards II


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Futures and Forwards IIFutures and Forwards II

In our second futures and forwards lecture we will:In our second futures and forwards lecture we will:

Develop pricing models for forwards when the underlying assetDevelop pricing models for forwards when the underlying assethas cash flows associated with it during the period of thehas cash flows associated with it during the period of theforward contract.forward contract.

Discuss the relationship between futures and forward prices.Discuss the relationship between futures and forward prices. Examine the relationship between forward/futures prices and theExamine the relationship between forward/futures prices and the

expected future spot price.expected future spot price.

Discuss accounting issues relating to hedging.Discuss accounting issues relating to hedging.


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Forward Contracts on a SecurityForward Contracts on a Security

that Provides a Known Cash Incomethat Provides a Known Cash Income Recall that we demonstrated that a forward contract on aRecall that we demonstrated that a forward contract on a

security that pays no dividends will have, at time 0, asecurity that pays no dividends will have, at time 0, a

delivery price of:delivery price of:

and that we could attribute this formula to the notion ofand that we could attribute this formula to the notion of

thethe opportunity costopportunity cost that the short party faced: to inducethat the short party faced: to inducethe short party to enter the contract, the long party mustthe short party to enter the contract, the long party mustpay them interest at least in the amount that the shortpay them interest at least in the amount that the shortcould get if they simply sold the asset now and invested itcould get if they simply sold the asset now and invested itat the riskat the risk--free rate.free rate.

0

rT F S e!


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that Provides a Known Cash Incomethat Provides a Known Cash Income Recall, however, that during the period of the forwardRecall, however, that during the period of the forward

contract, if the short party physically holds the underlyingcontract, if the short party physically holds the underlyingasset, then they will garner any benefits that accrue toasset, then they will garner any benefits that accrue tothe asset during that period.the asset during that period. For example, if the forward contract were written on a stock, andFor example, if the forward contract were written on a stock, and

the stock paid a dividend, then if the short party physically heldthe stock paid a dividend, then if the short party physically heldthe stock on thethe stock on the exex--dividenddividend date, they would receive thedate, they would receive thedividend.dividend.

The short party still has no risk (ignoring credit risk) inThe short party still has no risk (ignoring credit risk) in

the forward contract; as a result they should still onlythe forward contract; as a result they should still onlyearn the riskearn the risk--free rate for being the short party.free rate for being the short party.Consequently, the benefits that accrue to holding theConsequently, the benefits that accrue to holding theunderlying asset wouldunderlying asset would reducereduce the amount that the longthe amount that the longwould have to pay the short to induce them to enter thewould have to pay the short to induce them to enter the

contract.contract.


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that Provides a Known Cash Incomethat Provides a Known Cash Income To see this, consider a security that had a perfectlyTo see this, consider a security that had a perfectly

predicable set of cash flows that will occur between tpredicable set of cash flows that will occur between t

and T.and T. Examples are stocks that have known dividends and couponExamples are stocks that have known dividends and coupon

payments from bonds.payments from bonds.

Denote the present value of those cash flows as IDenote the present value of those cash flows as I

(discounting at the risk free rate). For their to be no(discounting at the risk free rate). For their to be noarbitrage, the relationship between F and S must be:arbitrage, the relationship between F and S must be:

F = (SF = (S--I)eI)erTrT


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Example:Example:

Assume a 10 month forward on a dividendAssume a 10 month forward on a dividend--paying stock. Currentpaying stock. Currentprice is $50. Assume r = 8%, and dividends of .75 in 3, 6, and 9price is $50. Assume r = 8%, and dividends of .75 in 3, 6, and 9months.months.

I = .75eI = .75e--.08(.25).08(.25) + .75e+ .75e--.08(.5).08(.5) + .75e+ .75e--.08(.75).08(.75) = 2.162= 2.162

TT--t = 10/12 = .83333 years:t = 10/12 = .83333 years:

F = (50F = (50--2.162)e2.162)e.08*.83333.08*.83333 = 51.136= 51.136


that Provides a Known Cash Incomethat Provides a Known Cash Income


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Second example:Second example:

Assume a 3 month forward on a dividendAssume a 3 month forward on a dividend--paying stock withpaying stock withcurrent price of $100. Assume r = 4%, and the stock will pay acurrent price of $100. Assume r = 4%, and the stock will pay adividend of $2.00 in 1 month.dividend of $2.00 in 1 month.

I = 2.00I = 2.00--.04(1/12).04(1/12) = 1.9933= 1.9933 TT--t = 3/12 = .25 years:t = 3/12 = .25 years:

F = (100F = (100--1.9933)e1.9933)e.04*.25.04*.25 = $98.99= $98.99

We can use this example to demonstrate the arbitrageWe can use this example to demonstrate the arbitrageopportunity that enforces this rule.opportunity that enforces this rule.




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To see this, consider what would be the case if this didTo see this, consider what would be the case if this didnot hold:not hold: Case 1: F > (SCase 1: F > (S00--I)eI)e

rTrT

Lets say that we saw F=101, how would the arbitrageur exploit theLets say that we saw F=101, how would the arbitrageur exploit theopportunity?opportunity?

At time 0:At time 0: Short the forward contract (i.e. agree to deliver the stock in threeShort the forward contract (i.e. agree to deliver the stock in three

months for $101).months for $101).

Borrow $100 today at the riskBorrow $100 today at the risk--free rate and buy the stock.free rate and buy the stock.

At time 1 month:At time 1 month: Reinvest the dividend at the riskReinvest the dividend at the risk--free rate.free rate.

At time 3 months they do 4 things:At time 3 months they do 4 things: Deliver the stock into the forward contract and receive $101.Deliver the stock into the forward contract and receive $101.

Receive $2.103 (2eReceive $2.103 (2e.04(2/12).04(2/12)) from the reinvested dividends.) from the reinvested dividends.

Repay $101.005 (100eRepay $101.005 (100e.04(3/12).04(3/12)) for the $100 you borrowed at time 0.) for the $100 you borrowed at time 0.

Net time 3 cash: +101 + 2.013Net time 3 cash: +101 + 2.013 101.005 = $2.008101.005 = $2.008




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We can show this on a timeline:We can show this on a timeline:



0 1 3

Actions

Cash Positions


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0 1 3

Actions

Cash Positions

1. Short futures contract

2. Borrow $100 at 4%3. Buy stock for $100

1. 02. +$1003. -$100

$0


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0 1 3

Actions

Cash Positions



1. 02. +$1003. -$100

$0

1. Receive $2 dividend

2.Reinvest at 4%

1. +$22. -$2

$0


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0 1 3

Actions

Cash Positions



1. 02. +$1003. -$100

$0


2.Reinvest at 4%

1. +$22. -$2

$0

1. Receive $101 from futures delivery

2.Repay loan ($101.005)3. Receive reinvested dividends (2.013)

1. +$101.0002. -$101.0053. -$ 2.013

+ 2.008


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What about the opposite situation?What about the opposite situation? Case 2: F < (SCase 2: F < (S00--I)eI)e

rTrT

Lets say that we saw F=98, how would the arbitrageur exploit theLets say that we saw F=98, how would the arbitrageur exploit theopportunity?opportunity?

At time 0:At time 0: Take a long position in the forward contract (i.e. agree to buy the stockTake a long position in the forward contract (i.e. agree to buy the stock

in three months at $98).in three months at $98).

Short the stock for $100 today and invest at riskShort the stock for $100 today and invest at risk--free rate.free rate.

At time 1 month:At time 1 month: Borrow $2 at risk free rate.Borrow $2 at risk free rate.

Pay $2 to the person from whom you borrowed the stock.Pay $2 to the person from whom you borrowed the stock. At time 3 months:At time 3 months:

Receive 101.005 (100eReceive 101.005 (100e.04(3/12).04(3/12)) from $100 you invested at risk) from $100 you invested at risk--free rate.free rate.

Buy stock for $98 via forward contract. Return stock to original owner.Buy stock for $98 via forward contract. Return stock to original owner.

Repay $2.013 (2eRepay $2.013 (2e.04(2/12).04(2/12)) for the $2 you borrowed at time 1.) for the $2 you borrowed at time 1.

Net time 3 cash: +101.005Net time 3 cash: +101.005 -- 2.0132.013-- 98.00 = $0.992098.00 = $0.9920




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We can also show this on a timeline:We can also show this on a timeline:



0 1 3

Actions

Cash Positions


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0 1 3

Actions

Cash Positions

1. Go long futures contract

2. Short stock at $1003. Invest $100 at 4%

1. 02. +$1003. -$100

$0


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0 1 3

Actions

Cash Positions



1. 02. +$1003. -$100

$0


2.Reinvest at 4%

1. +$22. -$2

$0


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0 1 3

Actions

Cash Positions



1. 02. +$1003. -$100

$0


2.Reinvest at 4%

1. +$22. -$2

$0

1. Receive $101 from futures delivery

2.Repay loan ($101.005)3. Receive reinvested dividends (2.013)

1. +$101.0002. -$101.0053. -$ 2.013

+ 2.008


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Forward Contracts on a Security thatForward Contracts on a Security that

Provides a Known Dividend YieldProvides a Known Dividend Yield Some securities such as stock indices and currenciesSome securities such as stock indices and currencies

essentially have a continuous dividend yield instead of aessentially have a continuous dividend yield instead of adiscrete dividend.discrete dividend.

Thus you can think of the asset as paying a continuousThus you can think of the asset as paying a continuousdividend at rate q, based on the value of the security.dividend at rate q, based on the value of the security. Thus if q=.10, and the security price is $50, the dividends in theThus if q=.10, and the security price is $50, the dividends in the

next small period of time are paid at the rate of $5 per year.next small period of time are paid at the rate of $5 per year.

The same basic logic for pricing forward contracts onThe same basic logic for pricing forward contracts oninstruments that pay a discrete dividend applies toinstruments that pay a discrete dividend applies toforward contracts on instruments that pay a dividendforward contracts on instruments that pay a dividendyield: the total earnings of the short party still should stillyield: the total earnings of the short party still should stillbe the riskbe the risk--free rate. As a result the dividend yieldfree rate. As a result the dividend yield

reducesreduces the rate that the long party must pay.the rate that the long party must pay.


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Provides a Known Dividend YieldProvides a Known Dividend Yield Thus, the formula for determining the forward price is:Thus, the formula for determining the forward price is:

F = SF = S00ee(r(r--q)Tq)T

where q is the dividend yield, expressed in annual terms.where q is the dividend yield, expressed in annual terms.

For example, if the S&P 500 had a dividend yield of 4%For example, if the S&P 500 had a dividend yield of 4%the sixthe six--month risk free rate were 3%, and the value ofmonth risk free rate were 3%, and the value ofthe S&P were 1009.37, then the 6 month forward pricethe S&P were 1009.37, then the 6 month forward price

for an S&P forward contract would be:for an S&P forward contract would be:F = 1009.37eF = 1009.37e(.03(.03--.04)(.5).04)(.5)= $1004.34= $1004.34

Note that when q>r, you will find anNote that when q>r, you will find an inverted marketinverted market. The next. The nextpage shows CME quotes for the S&P 500 index for 9/13/06.page shows CME quotes for the S&P 500 index for 9/13/06.


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Provides a Known Dividend YieldProvides a Known Dividend Yield

What does the fact that the prices are rising with maturity indicate?


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We can develop a somewhat more formal proof of theWe can develop a somewhat more formal proof of thepricing formula:pricing formula:

Again consider two portfolios,Again consider two portfolios,1.1. One long forward contract and cash equal to KeOne long forward contract and cash equal to Ke--r(Tr(T--t)t) (f+Ke(f+Ke--r(Tr(T--t)t)).).

2.2. 1*e1*e--q(Tq(T--t)t)

units of the security with all income being reinvested in the security. Thusunits of the security with all income being reinvested in the security. Thusat time T you will once again have one unit of the security, which is worth Sat time T you will once again have one unit of the security, which is worth STT..

Clearly Aand B once again have the same payoffs at time T, and soClearly Aand B once again have the same payoffs at time T, and soonce again, setting them equivalent:once again, setting them equivalent:

f+Kef+Ke--rTrT = Se= Se--qTqT

Or f = SeOr f = Se--qTqT -- KeKe--rTrT

setting f = 0 and solving for K leads to:setting f = 0 and solving for K leads to:

K = F = SeK = F = Se(r(r--q)Tq)T

Forward Contracts on a Security thatForward Contracts on a Security thatProvides a Known Dividend YieldProvides a Known Dividend Yield


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Forward Prices Versus Futures PricesForward Prices Versus Futures Prices

Hull demonstrates that if interest rates are constant, then forwards andHull demonstrates that if interest rates are constant, then forwards andfutures prices are the same.futures prices are the same.

Obviously in the real world interest rates are not constant, and so forwardObviously in the real world interest rates are not constant, and so forwardand futures prices are not the same.and futures prices are not the same.

The reason for this is the discounting of the mark to market cash flows.The reason for this is the discounting of the mark to market cash flows.

One way to see this is to consider if SOne way to see this is to consider if Stt is highly correlated with ris highly correlated with rtt-- so thatso thatwhen r increases, S tends to increase.when r increases, S tends to increase. When rates rise, the spot price of the asset will rise as well and so will the futuresWhen rates rise, the spot price of the asset will rise as well and so will the futures

price.price.

A short party in the futures contract will have to raise cash to mark to marketA short party in the futures contract will have to raise cash to mark to market when rates are high.when rates are high.

If rates fall, the spot price of the asset will fall, and so will the futures price. TheIf rates fall, the spot price of the asset will fall, and so will the futures price. The

short party will receive cash from the mark to market, when rates are low.short party will receive cash from the mark to market, when rates are low. So on average the short party in the futures contract must raise funds when ratesSo on average the short party in the futures contract must raise funds when rates

are high and invest funds when the rate is low. If instead they had used a forwardare high and invest funds when the rate is low. If instead they had used a forwardcontract they would not have any marking to marketcontract they would not have any marking to market so, assuming they wind upso, assuming they wind upat the same closing price, the short futures contract would be more expensiveat the same closing price, the short futures contract would be more expensivethan the short forward contract.than the short forward contract.

The short party, therefore, would demand a higher futures price to induce them toThe short party, therefore, would demand a higher futures price to induce them to

enter into this contract as opposed to a forward contract.enter into this contract as opposed to a forward contract.


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Forward Prices Versus Futures PricesForward Prices Versus Futures Prices

There are a some empirical papers that address theThere are a some empirical papers that address thedifferences between futures and forwards prices.differences between futures and forwards prices.

Cornell and Reinganum studied forward and futures prices onCornell and Reinganum studied forward and futures prices on

Pounds, C. Dollars, Marks, Yen and Swiss Francs and found littlePounds, C. Dollars, Marks, Yen and Swiss Francs and found littlestatistical differences between them.statistical differences between them.

For commodities and precious metals, French as well as ParkFor commodities and precious metals, French as well as Parkand Chen, find that forward and futures prices are significantlyand Chen, find that forward and futures prices are significantlydifferent from each other, and that the futures prices tend to bedifferent from each other, and that the futures prices tend to behigher than the forward prices.higher than the forward prices.


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Stock Index FuturesStock Index Futures

A stock index tracks the changes in the value of aA stock index tracks the changes in the value of ahypothetical portfolio of stocks. The weight of a stock inhypothetical portfolio of stocks. The weight of a stock inthe portfolio equals the proportion of the portfoliothe portfolio equals the proportion of the portfolioinvested in the stock.invested in the stock.

A few tidbits about stock index futures:A few tidbits about stock index futures:1.1. The percentage increase in the value of a stock index over aThe percentage increase in the value of a stock index over a

small interval of time is usually defined so that it is equal to thesmall interval of time is usually defined so that it is equal to thepercentage increase in the total value of the stocks in thepercentage increase in the total value of the stocks in theportfolio at that time.portfolio at that time.

2.2. Cash dividends received on the portfolio are ignored whenCash dividends received on the portfolio are ignored whenpercentage changes in most indices are calculated.percentage changes in most indices are calculated.

3.3. If the hypothetical portfolio of stocks remains fixed, the weightsIf the hypothetical portfolio of stocks remains fixed, the weightsassigned to each stock do not remain fixed. Consider thisassigned to each stock do not remain fixed. Consider thisexample of three stocks.example of three stocks.


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Stock IndicesStock Indices

In practice, the weights for stock indices are calculatedIn practice, the weights for stock indices are calculatedin one of two ways:in one of two ways:

Price weighted (DJIAand Nikkei 250): Stocks are pricePrice weighted (DJIAand Nikkei 250): Stocks are priceweightedweighted --> i.e. assume you own one share of each stock and> i.e. assume you own one share of each stock and

then add prices together (adjusting for splits, changes, etc.)then add prices together (adjusting for splits, changes, etc.) Market capitalization weighted (S&P and most other indices):Market capitalization weighted (S&P and most other indices):

weighted based on percentage of market capitalization forweighted based on percentage of market capitalization forentire portfolio:entire portfolio:

Stock Price * Shares Outstanding

Maket Capitalization of Entire PortfolioW

eight !


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Futures Prices of Stock IndicesFutures Prices of Stock Indices

As mentioned earlier, most indices can be thought of asAs mentioned earlier, most indices can be thought of asa security that pays dividends. Usually it is convenienta security that pays dividends. Usually it is convenientto consider them as paying dividends continuously. Letto consider them as paying dividends continuously. Letq be the dividend yield rate, then the futures price isq be the dividend yield rate, then the futures price is

given by:given by:

F = SeF = Se(r(r--q)(Tq)(T--T)T)..

If you were uncomfortable with using the dividend yieldIf you were uncomfortable with using the dividend yield

approach (say if you were using a small index), then you couldapproach (say if you were using a small index), then you couldestimate individual dividends and apply the formula for a futuresestimate individual dividends and apply the formula for a futurescontract on a security paying a known dividend.contract on a security paying a known dividend.


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Index ArbitrageIndex Arbitrage

Once again we can use arbitrage arguments to prove the equation:Once again we can use arbitrage arguments to prove the equation: If F > SeIf F > Se(r(r--q)(Tq)(T--T)T), then buy the stocks underlying the index and short, then buy the stocks underlying the index and short

futures contracts. If F < Sefutures contracts. If F < Se(r(r--q)(Tq)(T--T)T), do the opposite: short the stock and, do the opposite: short the stock andbuy the futures contract.buy the futures contract.

Because the transactions costs of buying the stocks are relativelyBecause the transactions costs of buying the stocks are relativelyhigh, arbitrage can sometimes be found in the market; this is knownhigh, arbitrage can sometimes be found in the market; this is knownas index arbitrage. Soas index arbitrage. So--called Program Trading is where yourcalled Program Trading is where yourcomputer system executes trades as soon as the arbitrage becomescomputer system executes trades as soon as the arbitrage becomespossible.possible.

Note that during the 1987 crash, there was a breakdown in theNote that during the 1987 crash, there was a breakdown in therelationship between F and S. This was due to poor mechanics ofrelationship between F and S. This was due to poor mechanics oftrading (on the 19trading (on the 19thth) and then due to restrictions placed by NYSE on) and then due to restrictions placed by NYSE onprogram trading (on the 20program trading (on the 20thth).).


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Hedging Using Index FuturesHedging Using Index Futures

Consider the problem faced by the manager of a largeConsider the problem faced by the manager of a largeequity portfolio:equity portfolio: They may wish to change the riskThey may wish to change the risk--profile of the portfolio, but doprofile of the portfolio, but do

not want to bear the transactions costs associated with changingnot want to bear the transactions costs associated with changingthe actual holdings.the actual holdings. Consider, for example, a large mutual fund such as FidelityConsider, for example, a large mutual fund such as Fidelity

Magellan, which has roughly $45 Billion in assets.Magellan, which has roughly $45 Billion in assets.**

It has a Beta of approximately 1.01 (relative to the S&P 500). So itIt has a Beta of approximately 1.01 (relative to the S&P 500). So ithas an almost perfect exposure to the general market risk of thehas an almost perfect exposure to the general market risk of thestock market.stock market.

Lets just assume for a moment that the managers of FidelityLets just assume for a moment that the managers of Fidelitydecided that they wanted to reduce the exposure of the fund todecided that they wanted to reduce the exposure of the fund tothe stock market (assuming that the prospectus of the fundthe stock market (assuming that the prospectus of the fundwould allow them to do this.) Lets say they want to reduce thewould allow them to do this.) Lets say they want to reduce theBeta of the portfolio to 0.5.Beta of the portfolio to 0.5.

*As of 9/12/2003. Source: Fidelity web site (http://personal.fidelity.com/products/funds/mfl_frame.shtml?316184100 )


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One way to do this would be to sell half of the funds assets andOne way to do this would be to sell half of the funds assets andinvest it in the riskinvest it in the risk--free rate.free rate. This is not practical, there is no way they could unload $22.5 BillionThis is not practical, there is no way they could unload $22.5 Billion

in stocks without depressing the market. They would get terriblein stocks without depressing the market. They would get terribleprices for the assets, not to mention that they would lose a lot ofprices for the assets, not to mention that they would lose a lot of

money on commissions as well. Further, selling the assets wouldmoney on commissions as well. Further, selling the assets wouldtake quite a bit of time.take quite a bit of time.

In addition, the fund managers may only want toIn addition, the fund managers may only want to temporarilytemporarilychange the risk profile of the portfolio. They may want to hold thechange the risk profile of the portfolio. They may want to hold thestocks for the long run, but want to temporarily reduce the risk ofstocks for the long run, but want to temporarily reduce the risk ofthe portfoliothe portfolio for say 3 months.for say 3 months.

A relatively easy way for them to change the portfolios riskA relatively easy way for them to change the portfolios riskwould be to hedge that risk using a stock index futures contract.would be to hedge that risk using a stock index futures contract.

The next few slides discuss the general method for doing thisThe next few slides discuss the general method for doing this(as outlined in Hulls book), and then we will return to an(as outlined in Hulls book), and then we will return to anexample based on Magellan.example based on Magellan.


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Recall the definition ofRecall the definition of: it is the slope of the regression: it is the slope of the regressionline between the excess returns (i.e. returns less the riskline between the excess returns (i.e. returns less the riskfree rate) on a portfolio of stocks and the excess returnfree rate) on a portfolio of stocks and the excess returnon the market (i.e. the stock market as a whole.)on the market (i.e. the stock market as a whole.)

WhenWhen =1, the return on the portfolio tends to mirror the return=1, the return on the portfolio tends to mirror the returnon the market: i.e. the portfolio has the same risk as the market.on the market: i.e. the portfolio has the same risk as the market.WhenWhen =2, the excess return on the portfolio is twice that of the=2, the excess return on the portfolio is twice that of themarketmarket -- its risk is twice as high.its risk is twice as high.

ConsiderConsider also that for a futures contract on large indices, suchalso that for a futures contract on large indices, suchas the S&P 500, the index can serve as a proxy for the marketas the S&P 500, the index can serve as a proxy for the marketas a whole.as a whole. Thus we can say that if a portfolio has aThus we can say that if a portfolio has a =3, its expected excess=3, its expected excess

return will be equal to three times the excess return on thereturn will be equal to three times the excess return on theunderlying index.underlying index.

t t( ort olio Return ) ( arket Return )t tr r eE F


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Hull defines the following terms:Hull defines the following terms:

PP -- the value of the portfoliothe value of the portfolio

AA -- underlying asset value of one futures contract (if one futuresunderlying asset value of one futures contract (if one futurescontract is on m times the index, A= mF)contract is on m times the index, A= mF)

If you are working with a portfolio that exactly mirrors aIf you are working with a portfolio that exactly mirrors aspecific index upon which futures contracts are written,specific index upon which futures contracts are written,and you want to completely hedge the risk of theand you want to completely hedge the risk of theportfolio, all you have to do is short an appropriateportfolio, all you have to do is short an appropriate

number of future contracts.number of future contracts. The optimal number of contracts to short when hedging is:The optimal number of contracts to short when hedging is:

NN** = P/A= P/A


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If, however, your portfolio does not exactly match theIf, however, your portfolio does not exactly match theunderlying index, (but assuming that your underlyingunderlying index, (but assuming that your underlyingindex has a Beta of 1), then the optimal number ofindex has a Beta of 1), then the optimal number offutures contracts to completely hedge the risk isfutures contracts to completely hedge the risk is

proportional to the Beta of your portfolio:proportional to the Beta of your portfolio:

NN** == (P/A)(P/A)

Note that if the your index had a Beta that wasNote that if the your index had a Beta that wassomething other than one, the optimal number futuressomething other than one, the optimal number futurescontracts would be proportional to the ratio of yourcontracts would be proportional to the ratio of yourportfolios Beta to the Beta of the underlying index, withportfolios Beta to the Beta of the underlying index, withboth of those Betas defined relative to the sameboth of those Betas defined relative to the sameunderlying market.underlying market.


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The previous formulas, while important, do not exactly help ourThe previous formulas, while important, do not exactly help ourMagellan fund manager. They do not want to completely eliminateMagellan fund manager. They do not want to completely eliminatethe exposure to the stock market (i.e. to have a netthe exposure to the stock market (i.e. to have a net of 0), butof 0), butrather want to reduce that exposure in half, i.e. to have a netrather want to reduce that exposure in half, i.e. to have a net ofof0.5.0.5.

LetLet be the current portfolio position and letbe the current portfolio position and let** be the desired portfoliobe the desired portfolioposition. To get to the new portfolio position the optimal number ofposition. To get to the new portfolio position the optimal number ofindex futures contracts to short is:index futures contracts to short is:

NN**= (= (-- **)(P/A))(P/A)

Note that ifNote that if**>> , that is, you want to increase risk, you will get a, that is, you want to increase risk, you will get anegative number for Nnegative number for N**, that means you take a LONG position (since N, that means you take a LONG position (since N**

is the number of contracts to short). Hull flipsis the number of contracts to short). Hull flips andand ** and then saysand then saysthat is the number of contract to go long, but the effect is the same!that is the number of contract to go long, but the effect is the same!


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So let us return to our Magellan example.So let us return to our Magellan example. Remember that theRemember that the of Magellan is 1.01, and that we want toof Magellan is 1.01, and that we want to

reduce it to 0.50.reduce it to 0.50.

We will assume that we want to reduce that risk for 1 month.We will assume that we want to reduce that risk for 1 month.

We should use the S&P 500 index futures contract to do this.We should use the S&P 500 index futures contract to do this. There are two S&P 500 Index Futures contracts, and each trade onThere are two S&P 500 Index Futures contracts, and each trade on

the CME.the CME.

The S&P Index Futures contract is based on $250 times the indexThe S&P Index Futures contract is based on $250 times the indexamount.amount.

The Mini S&P Futures contract is based on $20 times the indexThe Mini S&P Futures contract is based on $20 times the index

amount.amount.

Obviously to hedge a $45 Billion position, we will use the largerObviously to hedge a $45 Billion position, we will use the largercontract.contract.


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Hedging Using Index FuturesHedging Using Index Futures On 9/12/2006, the S&P 500 settled at 1318.07, and the S&P 500 indexOn 9/12/2006, the S&P 500 settled at 1318.07, and the S&P 500 index

futures contract settled atfutures contract settled at1317.601317.60. The value of the portfolio was. The value of the portfolio was(roughly) $45,000,000,000. Thus:(roughly) $45,000,000,000. Thus:

P = $45,000,000,000P = $45,000,000,000

A= $250*1317.60 = 329,400.00A= $250*1317.60 = 329,400.00

= 1.01= 1.01

**= 0.50= 0.50

So our optimal number of contracts to hedge would be:So our optimal number of contracts to hedge would be:

NN**=(=(--**)*(P/A))*(P/A)

NN**=(1.01=(1.01--0.50)*(45,000,000,000/329,400)0.50)*(45,000,000,000/329,400)

NN**

=(0.51)*(136,612.02) = 69,672 contracts.=(0.51)*(136,612.02) = 69,672 contracts.

In reality, there will be a problem with this, because the largest position theIn reality, there will be a problem with this, because the largest position theCME allows is 20,000 contracts in a given month, so you would really haveCME allows is 20,000 contracts in a given month, so you would really haveto spread your hedge over 6 or 7 months worth of contracts, but we willto spread your hedge over 6 or 7 months worth of contracts, but we willignore this for now.ignore this for now.


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So how well would the hedge work?So how well would the hedge work? On 9/13/2003, the S&P is at 1318.07, and the net value of the portfolioOn 9/13/2003, the S&P is at 1318.07, and the net value of the portfolio

was $45,000,000,000.was $45,000,000,000.

Assume that on 9/18/2006, the S&P closes at 1351.09, so it hadAssume that on 9/18/2006, the S&P closes at 1351.09, so it had

increased in value by 2.505%. Since theincreased in value by 2.505%. Since the of the portfolio was 1.01, weof the portfolio was 1.01, wewould expect the value of the portfolio to increase by 2.505*1.01, orwould expect the value of the portfolio to increase by 2.505*1.01, or2.53%. Thus, the new portfolio value would be:2.53%. Thus, the new portfolio value would be:

45,000,000,000 * 1.0253 = 46,127,250,000.45,000,000,000 * 1.0253 = 46,127,250,000.That is you have made (46,127,250,000That is you have made (46,127,250,000--45,000,000,000 = 1,127,250,000)45,000,000,000 = 1,127,250,000)on the portfolio.on the portfolio.

Of course, the S&P index futures would rise as well, and you areOf course, the S&P index futures would rise as well, and you are shortshortthe index. Indeed, assume that the September Futures contract onthe index. Indeed, assume that the September Futures contract on9/18/2003 settles at 1349.61. So your hedge has moved:9/18/2003 settles at 1349.61. So your hedge has moved:

--69,672 * (1349.6169,672 * (1349.61--1317.60)*250 =1317.60)*250 = --557,479,467557,479,467


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So what is the net change in your position?So what is the net change in your position?

Remember, you made 1,127,250,000 on the portfolio, but lostRemember, you made 1,127,250,000 on the portfolio, but lost557,479,467 on the hedge (the S&P index futures), so net you557,479,467 on the hedge (the S&P index futures), so net youhave made: 1,127,250,000have made: 1,127,250,000 -- 557,479,467 = 569,770,533 on557,479,467 = 569,770,533 on

your net position.your net position. This is 50.5% of what you would have made had you notThis is 50.5% of what you would have made had you not

hedged.hedged.

Why was it not exactly 50% or 49%? Because theWhy was it not exactly 50% or 49%? Because the of yourof yourportfolio was probably changing when rates change.portfolio was probably changing when rates change.

This illustrates the notion that you will frequently makeThis illustrates the notion that you will frequently makemore money by not hedging, but that what you aremore money by not hedging, but that what you aredoing is reducing risk.doing is reducing risk.


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Indeed, the following table shows the hedged and unIndeed, the following table shows the hedged and un--hedged position for various dates in September, 2006.hedged position for various dates in September, 2006.

Date S&P Magellan's Portfolio Index Futures Change in Hedge Net Value, Hedged Net Value, No Hedge

8/29/2006 1,304.28 44,524,600,564.11 1305.10 44,524,600,564.11 44,524,600,564.11

9/5/2006 1,313.25 44,833,874,281.21 1314.60 (165,471,000.00) 44,668,403,281.21 44,833,874,281.21

9/8/2006 1,298.92 44,339,761,117.20 1299.3 266,495,400.00 44,606,256,517.20 44,339,761,117.209/12/2006 1,313.00 44,825,200,462.88 1313.7 (250,819,200.00) 44,574,381,262.88 44,825,200,462.88

Hedged vs. Unhedegd Positions of Portfolio

44.30

44.40

44.50

44.60

44.70

44.80

44.90

8/27/20

06

8/29/20

06

8/31/20

06

9/2/200

6

9/4/200

6

9/6/200

6

9/8/200

6

9/10/20

06

9/12/20

06

9/14/20

06

(billions)

Date

Value

Net Value, Hedged Net Value, No Hedge


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Hedging an Individual Stocks exposure.Hedging an Individual Stocks exposure. Basically you just treat the stock as a portfolio, and use itsBasically you just treat the stock as a portfolio, and use its toto

determine the hedge ratio.determine the hedge ratio.

Forwards and Futures on CurrenciesForwards and Futures on Currencies

You are not responsible for Forwards/Futures on CurrenciesYou are not responsible for Forwards/Futures on Currencies


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Commodities FuturesCommodities Futures

Commodities futures contracts fall into two groups:Commodities futures contracts fall into two groups:those where the underlying is held primarily forthose where the underlying is held primarily forinvestment and those where the underlying is heldinvestment and those where the underlying is heldprimarily for consumption. This is really importantprimarily for consumption. This is really important

because if it is held for consumption some of ourbecause if it is held for consumption some of ourarbitrage arguments break down and no longer hold.arbitrage arguments break down and no longer hold.

Examples: Gold is held primarily for investment, whereas wheat,Examples: Gold is held primarily for investment, whereas wheat,oil, cattle, etc. is primarily held for consumption.oil, cattle, etc. is primarily held for consumption.


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Commodities Held for InvestmentCommodities Held for Investment

Some commodities, primarily gold and silver, are held bySome commodities, primarily gold and silver, are held bya significant number of investors solely for investment.a significant number of investors solely for investment.If storage costs are zero, then they can be treated asIf storage costs are zero, then they can be treated assecurities that pay no income. Thus the correct formulasecurities that pay no income. Thus the correct formula

is:is:F = SeF = Ser(Tr(T--t)t)

If there is a fixed storage cost, then let U be the presentIf there is a fixed storage cost, then let U be the presentvalue of those storage costs incurred during the life ofvalue of those storage costs incurred during the life of

the contract, and treat it as negative income. Thisthe contract, and treat it as negative income. Thisallows us to use the standard formula for securities thatallows us to use the standard formula for securities thatpay a known income:pay a known income:

F = (SF = (S -- ((--U))eU))er(Tr(T--t)t) = (S+U)e= (S+U)er(Tr(T--t)t)


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If the storage costs are proportional to the price of theIf the storage costs are proportional to the price of thecommodity, they can be treated as a negative dividendcommodity, they can be treated as a negative dividendyield:yield:

F = SeF = Se(r(r--((--u))(Tu))(T--t)t) =Se=Se(r+u)(T(r+u)(T--t)t)

where u is a per annum proportion of the spot price.where u is a per annum proportion of the spot price.

Consider a one year gold futures contract, and assume that itConsider a one year gold futures contract, and assume that itcosts $2 per ounce to store gold, payment made at the end ofcosts $2 per ounce to store gold, payment made at the end of

the year. If the spot price is $450 and the risk free rate is 7%the year. If the spot price is $450 and the risk free rate is 7%for one year, then:for one year, then:

U = 2.00eU = 2.00e--.07.07=1.865, and so the forward price would be:=1.865, and so the forward price would be:

F = (SF = (S00+U)e+U)erTrT = (450+1.865)e= (450+1.865)e.07(1).07(1)=484.63=484.63


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Suppose that an arbitrageur observes a violation of theSuppose that an arbitrageur observes a violation of thepricing rule of this type:pricing rule of this type:

F>(S+U)eF>(S+U)er(Tr(T--t)t)

A

s usual, they would take the following actions toA

s usual, they would take the following actions toexploit the arbitrage:exploit the arbitrage:1.1. Borrow S+U dollars at the riskBorrow S+U dollars at the risk--free rate and use it to buy onefree rate and use it to buy one

unit of the commodity and to pay the storage costs.unit of the commodity and to pay the storage costs.

2.2. Short a futures contract on the commodity.Short a futures contract on the commodity.

Clearly, then, as arbitrageurs do this F will approachClearly, then, as arbitrageurs do this F will approach(S+U)e(S+U)er(Tr(T--t)t), so the situation will not last long., so the situation will not last long.


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Commodities Held for InvestmentCommodities Held for Investment Suppose next that the arbitrageur notes the following:Suppose next that the arbitrageur notes the following:

F


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Convenience YieldsConvenience Yields

In the equation above, the difference between F andIn the equation above, the difference between F and(S+U)e(S+U)er(Tr(T--t)t) is theis the convenience yieldconvenience yield. Frequently it can. Frequently it canbe written as:be written as:

FeFey(Ty(T--t)t) = (S+U)e= (S+U)er(Tr(T--t)t)

where y is the c. yield. If using the proportional version:where y is the c. yield. If using the proportional version:

FeFey(Ty(T--t)t) = Se= Se(r+u)(T(r+u)(T--t)t)

or rewriting this in the more usual form:or rewriting this in the more usual form:

F = SeF = Se(r+u(r+u--y)(Ty)(T--t)t)


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The General Cost of Carry ModelThe General Cost of Carry Model

All of the futures and spot price relationships we have seen beforeAll of the futures and spot price relationships we have seen beforehave a common theme. The general name for this is the cost ofhave a common theme. The general name for this is the cost ofcarry model. Essentially this says that the futures price is acarry model. Essentially this says that the futures price is afunction of the cost of carrying (holding) the underlying asset. Thisfunction of the cost of carrying (holding) the underlying asset. Thisis consistent with our intuition that the forward price is really a formis consistent with our intuition that the forward price is really a form

of compensating theof compensating the shortshort for holding the asset for the longfor holding the asset for the longparty.party.

For nonFor non--dividend paying securities, the cost of carry is r.dividend paying securities, the cost of carry is r.

For a stock index it is rFor a stock index it is r--q, where q is the dividend yield.q, where q is the dividend yield.

For a commodity with storage costs proportional to its price, it is r+u.For a commodity with storage costs proportional to its price, it is r+u.

Define the cost of carry as c. The general model to use, then is:Define the cost of carry as c. The general model to use, then is:

F = S eF = S ec(Tc(T--t)t)

and for a consumption asset it is:and for a consumption asset it is:

F = S eF = S e(c(c--y)(Ty)(T--t)t)..


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Futures Pricing ModelsFutures Pricing Models

Weve developed quite a few pricing models, lets makeWeve developed quite a few pricing models, lets makesure we enunciate them:sure we enunciate them:

Security paying no income: F = SSecurity paying no income: F = S00eerTrT

Security with known discrete dividends: F = (SSecurity with known discrete dividends: F = (S00--I)eI)erTrT

Security with dividend yield: F = SSecurity with dividend yield: F = S00ee(r(r--y)Ty)T

Investment Commodity with fixed storage costs: F = (SInvestment Commodity with fixed storage costs: F = (S00+U)e+U)erTrT

Investment Commodity with proportional storage costs: F=SInvestment Commodity with proportional storage costs: F=S00ee(r+u)T(r+u)T

Consumption Commodity with fixed storage costs: F


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Delivery ChoicesDelivery Choices

Most futures contracts do not specify a delivery date, butMost futures contracts do not specify a delivery date, butrather a range when delivery is permissible. As a result,rather a range when delivery is permissible. As a result,the short party in the contract will choose to makethe short party in the contract will choose to makedelivery when it is to their advantage. This presents adelivery when it is to their advantage. This presents a

problem when trying to determine futures prices: whatproblem when trying to determine futures prices: whatday to use for T in all of the (Tday to use for T in all of the (T--t) equations?t) equations?

Consider the dividend yield model: F=SConsider the dividend yield model: F=S00ee(r(r--y)Ty)T..

If F increases as T increases, the benefits from holding the asset areIf F increases as T increases, the benefits from holding the asset areless than the riskless than the risk--free ratefree rate -- so the short party must want to get theirso the short party must want to get their

money out as soon as possible. They will then want to make deliverymoney out as soon as possible. They will then want to make deliveryas soon as possible, so you should assume T will be the soonest dateas soon as possible, so you should assume T will be the soonest datepossible.possible.

If, however, F decreases as T increases, the opposite is true, and theyIf, however, F decreases as T increases, the opposite is true, and theywill deliver as late as possible, or so you should assume.will deliver as late as possible, or so you should assume.


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4949

Futures Prices and the ExpectedFutures Prices and the Expected

Future Spot PriceFuture Spot Price A common question is whether the futures price is theA common question is whether the futures price is the

same as the expected future spot price. That is, is thesame as the expected future spot price. That is, is theprice of a wheat futures contract for delivery in threeprice of a wheat futures contract for delivery in three

months the same as the markets expected spot price ofmonths the same as the markets expected spot price ofthat wheat in three months?that wheat in three months?

I would begin by pointing out that none of our pricingI would begin by pointing out that none of our pricing

equations utilize the expected spot price, so I would notequations utilize the expected spot price, so I would notexpect any such relationship to hold.expect any such relationship to hold.


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5050

Futures Prices and the ExpectedFutures Prices and the ExpectedFuture Spot PriceFuture Spot Price

To answer this we must look at a risk and returnTo answer this we must look at a risk and returnanalysis.analysis.

From CAPM, recall that there are two types of risk. NonFrom CAPM, recall that there are two types of risk. Non--systematic risk is that risk which can be diversified away,systematic risk is that risk which can be diversified away,

whereas systematic risk cannot. In general, the higher thewhereas systematic risk cannot. In general, the higher thesystematic risk of an investment, the higher the expected returnsystematic risk of an investment, the higher the expected returndemanded by an investor.demanded by an investor.

Consider now the risk in a futures position.Consider now the risk in a futures position.

Begin by assuming that at time t the speculator puts the presentBegin by assuming that at time t the speculator puts the present

value of the futures price into a riskvalue of the futures price into a risk--free investment, i.e. theyfree investment, i.e. theyinvest:invest:

FeFe--rTrT

at the risk free rate (giving F dollars at time T) and they take aat the risk free rate (giving F dollars at time T) and they take along position in the futures contract.long position in the futures contract.


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At time T they use the cash in the riskAt time T they use the cash in the risk--free account (F) to takefree account (F) to takedelivery of the contract and then sell the commodity for Sdelivery of the contract and then sell the commodity for Sdollars.dollars.

Thus the cash flows are:Thus the cash flows are:

Time 0:Time 0: --FeFe--r(Tr(T--t)t)

Time T: +STime T: +STT

At time 0 the present value of the investment is:At time 0 the present value of the investment is:

--FeFe--r(Tr(T--t)t) + E(S+ E(STT

)e)e--k(Tk(T--t)t)

where k is the discount rate for this stock.where k is the discount rate for this stock.


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Note thatNote that --FeFe--r(Tr(T--t)t) is an amount, not a discount rate.is an amount, not a discount rate.

Assuming that all investment opportunities in securitiesAssuming that all investment opportunities in securitiesmarkets havemarkets have zero net present valuezero net present value::

--FeFe--r(Tr(T--t)t) + E(S+ E(STT)e)e--k(Tk(T--t)t) = 0, or= 0, or

F = E(SF = E(STT)e)e(r(r--k)(Tk)(T--t)t)

Thus only if k=r will F=E(SThus only if k=r will F=E(STT).).

What does it mean that all investment opportunities in aWhat does it mean that all investment opportunities in asecurities market are correctly priced: you do not earn (onsecurities market are correctly priced: you do not earn (onaverage) more than the appropriate riskaverage) more than the appropriate risk--adjusted rate for thatadjusted rate for thatasset, i.e. there are no arbitrage opportunities.asset, i.e. there are no arbitrage opportunities.


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5353

General ResultGeneral Result

Forward contracts, by design, are set to have an initialForward contracts, by design, are set to have an initialvalue, fvalue, ftt= 0. As time progresses, this value may become= 0. As time progresses, this value may becomepositive or negative.positive or negative.

There is a general relationship, applicable to all forwardThere is a general relationship, applicable to all forwardcontracts, that gives the value of a long road contract, f, incontracts, that gives the value of a long road contract, f, interms of the originally negotiated delivery price K and theterms of the originally negotiated delivery price K and thecurrent forward price F:current forward price F:

f = (Ff = (F--K)eK)e--r(Tr(T--t)t)

Note that if you compare two forwards, one with deliveryNote that if you compare two forwards, one with deliveryset at K and one at F, but otherwise identical, then the onlyset at K and one at F, but otherwise identical, then the onlydifference between the two is the delivery price.difference between the two is the delivery price.Discounting that back to today yields the above formula.Discounting that back to today yields the above formula.


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SeasonalitySeasonality

Certain commodities, especially agricultural and fuelCertain commodities, especially agricultural and fuelcommodities, have a seasonal component.commodities, have a seasonal component.

Corn, for example, is harvested in the fall but is consumedCorn, for example, is harvested in the fall but is consumedthroughout the year.throughout the year.

If corn supply/demand were constant, then you would expectIf corn supply/demand were constant, then you would expectthe price at harvest to be the same year after year.the price at harvest to be the same year after year.

Of course since it is consumed throughout the year it requiresOf course since it is consumed throughout the year it requiresstorage, so you would expect to see futures prices rising throughstorage, so you would expect to see futures prices rising throughthe year at the risk free rate plus storage costs (i.e. F=Sthe year at the risk free rate plus storage costs (i.e. F=S00ee

(r(r--u)Tu)T).).

The difference here is that u changes, essentially it is lower forThe difference here is that u changes, essentially it is lower formaturities near the harvest and larger for maturities further frommaturities near the harvest and larger for maturities further fromthe harvest.the harvest.


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Futures Hedge AccountingFutures Hedge Accounting

Financial Account Standard (FAS) #133, specifies howFinancial Account Standard (FAS) #133, specifies howfirms that use futures contracts must account for thefirms that use futures contracts must account for thehedge transactions.hedge transactions.

These standards primarily affect two of the firmsThese standards primarily affect two of the firms

accounting statements: the balance sheet and theaccounting statements: the balance sheet and theincome statement.income statement.

The balance sheet lists the value of each asset and liability ofThe balance sheet lists the value of each asset and liability ofthe firm.the firm.

Many items, such as inventory, are recorded on the balance sheetMany items, such as inventory, are recorded on the balance sheetat their purchase price, regardless of their current market value.at their purchase price, regardless of their current market value.These values are known as book value and may not be related toThese values are known as book value and may not be related tocurrent market values. This is also true for certain liabilities that thecurrent market values. This is also true for certain liabilities that thefirm may have on their books.firm may have on their books.


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The income statement is intended to show the netThe income statement is intended to show the netincome (i.e. earnings) of the firm.income (i.e. earnings) of the firm.

Traditionally items such as inventory did not show up in theTraditionally items such as inventory did not show up in theincome statement except when it was purchased and then whenincome statement except when it was purchased and then when

it was sold.it was sold. Speculative futures positions have to be shown on bothSpeculative futures positions have to be shown on both

the balance sheet and the income statement at their fairthe balance sheet and the income statement at their fairmarket value as of the reporting date.market value as of the reporting date.

The problem was (and still is to some degree) that when you putThe problem was (and still is to some degree) that when you put

a hedge on with a financial derivative, the change in thea hedge on with a financial derivative, the change in thederivative would show up in the income statement, but thederivative would show up in the income statement, but thechange in the asset you were hedging (if say inventory) wouldchange in the asset you were hedging (if say inventory) wouldnot! The accounting system tended to only show half the hedge.not! The accounting system tended to only show half the hedge.


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As more firms began to use derivatives to hedge, itAs more firms began to use derivatives to hedge, itbecame clear that the system had to be changed. Thebecame clear that the system had to be changed. Theidea behind FAS 133 was to try to modernize theidea behind FAS 133 was to try to modernize theaccounting system.accounting system.

FAS 133 allows futures positions used for hedging to beFAS 133 allows futures positions used for hedging to beaccounted for differently than speculative positions.accounted for differently than speculative positions.

The full text and a summary of FAS 133 can be found atThe full text and a summary of FAS 133 can be found atthis web site:this web site:

http://www.fasb.org/st/#fas50http://www.fasb.org/st/#fas50


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FAS 133FAS 133

General RuleGeneral Rule

The general rule is that all derivatives must be markedThe general rule is that all derivatives must be marked--toto--market and recorded on the companys balance sheet asmarket and recorded on the companys balance sheet asseparate assets/liabilities each period.separate assets/liabilities each period.

Derivatives that are used for hedgingDerivatives that are used for hedging maymay qualify for hedgequalify for hedgeaccounting treatment, if specific criteria are met.accounting treatment, if specific criteria are met.

FAS 133 covers three different types of hedges that a firm mightFAS 133 covers three different types of hedges that a firm mightuse:use:

1.1. Cash Flow HedgeCash Flow Hedge

2.2. Fair ValueH

edgeFair ValueH

edge3.3. Foreign Currency HedgeForeign Currency Hedge


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FAS 133FAS 133

Cash Flow HedgeCash Flow Hedge Aderivative that hedges the changes inAderivative that hedges the changes inforecasted cash flows of a transactionforecasted cash flows of a transaction

FairFair--Value HedgeValue Hedge Aderivative that hedges the changes in itsAderivative that hedges the changes in itsfair value of underlying assets or liabilities.fair value of underlying assets or liabilities.

A cashA cash--flow hedge must be a hedge instigated to protect againstflow hedge must be a hedge instigated to protect againstprice changes in a specific, identified underlying instrument. Theprice changes in a specific, identified underlying instrument. Thenet effect is that gains/losses on the hedge position are deferrednet effect is that gains/losses on the hedge position are deferreduntil the forecasted cash flows occur.until the forecasted cash flows occur.

A fairA fair--value hedge is a hedge instigated to hedge the pricevalue hedge is a hedge instigated to hedge the pricechange in an asset or liability. Both the hedge instrument (i.e.change in an asset or liability. Both the hedge instrument (i.e.the futures contract) and the hedged instrument/asset arethe futures contract) and the hedged instrument/asset aremarked to market each period.marked to market each period.


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FAS 133FAS 133

Here are the summaries of each of these:Here are the summaries of each of these:

Cash flow hedge:Cash flow hedge:

For a derivative designated as hedging the exposure to variable cash flowsFor a derivative designated as hedging the exposure to variable cash flowsof a forecasted transaction (referred to as a cash flow hedge), the effectiveof a forecasted transaction (referred to as a cash flow hedge), the effectiveportion of the derivative's gain or loss is initially reported as a component ofportion of the derivative's gain or loss is initially reported as a component of

other comprehensive income (outside earnings) and subsequentlyother comprehensive income (outside earnings) and subsequentlyreclassified into earnings when the forecasted transaction affects earnings.reclassified into earnings when the forecasted transaction affects earnings.The ineffective portion of the gain or loss is reported in earningsThe ineffective portion of the gain or loss is reported in earningsimmediately.immediately.

Fair Value hedge:Fair Value hedge:

For a derivative designated as hedging the exposure to changes in the fairFor a derivative designated as hedging the exposure to changes in the fairvalue of a recognized asset or liability or a firm commitment (referred to asvalue of a recognized asset or liability or a firm commitment (referred to as

a fair value hedge), the gain or loss is recognized in earnings in the perioda fair value hedge), the gain or loss is recognized in earnings in the periodof change together with the offsetting loss or gain on the hedged itemof change together with the offsetting loss or gain on the hedged itemattributable to the risk being hedged. The effect of that accounting is toattributable to the risk being hedged. The effect of that accounting is toreflect in earnings the extent to which the hedge is not effective in achievingreflect in earnings the extent to which the hedge is not effective in achievingoffsetting changes in fair value.offsetting changes in fair value.


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FAS 133FAS 133

The real difficulty with FAS 133 is that it applies only toThe real difficulty with FAS 133 is that it applies only tohighly effective hedges. Meaning that you musthighly effective hedges. Meaning that you mustdemonstrate (quantitatively) how effective a derivativedemonstrate (quantitatively) how effective a derivativewill be at hedging a risk.will be at hedging a risk.

Generally this has been held to mean that the hedge instrumentGenerally this has been held to mean that the hedge instrumentmust produce the inverse of 80% to 125% of the cumulativemust produce the inverse of 80% to 125% of the cumulativechange in the value of the instrument being hedged.change in the value of the instrument being hedged.

To the degree that the hedge is not highly effective, you haveTo the degree that the hedge is not highly effective, you haveto allocate part of the hedge instruments changes to the hedge,to allocate part of the hedge instruments changes to the hedge,

and part to hedge ineffectiveness.and part to hedge ineffectiveness. The hedge instrument must be highly effective when you firstThe hedge instrument must be highly effective when you first

put it onput it on and throughoutthe life of the hedgeand throughoutthe life of the hedge. Literally you. Literally youcan lose your right to hedge accounting.can lose your right to hedge accounting.


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6262

FAS 133FAS 133

Thus, you must constantly demonstrate that your hedgeThus, you must constantly demonstrate that your hedgeis highly effective.is highly effective.

This is not too bad if the underlying instrument you are hedgingThis is not too bad if the underlying instrument you are hedgingis liquid, and there is an easy to define market price for it, but itis liquid, and there is an easy to define market price for it, but it

can become a very big problem if the underlying is not thatcan become a very big problem if the underlying is not thatliquid; if there is not a lot of trading in the instrument, how doliquid; if there is not a lot of trading in the instrument, how doyou prove the effectiveness of the hedges?you prove the effectiveness of the hedges?

Mortgage Servicing Rights are a good example of this.Mortgage Servicing Rights are a good example of this.


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Futures and ForwardsFutures and Forwards

This ends our discussion of futures and forwardThis ends our discussion of futures and forwardcontracts.contracts.

We next move on to options contracts, although we willWe next move on to options contracts, although we willcontinue to use forwards both in deriving option andcontinue to use forwards both in deriving option and

other derivatives pricing formulas, and, of course, whenother derivatives pricing formulas, and, of course, whenvaluing options on futures contracts.valuing options on futures contracts.

Topic 4 Futures Pricing II

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Transcript of Topic 4 Futures Pricing II