FUTURES

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FUTURES

• DEFINITION
• Futures (forward) contracts are agreements
  between two agents where one agrees to purchase
  and the other to sell (deliver) a given amount
  of a specific commodity at a specific price at a
  future (prompt or delivery) date.
• Like an order for furniture, house, car, etc. at
  fixed
• price.
• BASIC FEATURES
• Both parties are "obliged" -gt not optional
• Buy (long) - sell (short)
• Like options - zero sum - derivative security
• No money changes hands initially - Margin put up
• Mark to market daily - money moves between

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FUTURES VS. FORWARD CONTRACTS

• FUTURES CONTRACTS ARE STANDARDIZED
• sold on exchanges - Chicago Board Trade (1848)
• involve clearing house
• trading in pit - no specialist - different
  prices
• FORWARD CONTRACTS ARE NOT STANDARDIZED
• sold over the phone (over the counter)
• no clearing house
• common for currency trading - banks
• 24 hour market
• no mark to market - end day settlement
• allow contingent delivery - sale of house etc.
• Look at Futures Quotes www.futures.quote.com
• QUESTION Which commodities are likely to have
• futures traded? (A commodity that can be
  graded - standardized - widely used - volatile
  price)

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1. In order to simplify things and focus on the most
   important issues, I will assume futures and
   forward prices are the same.
2. When the risk-free rate and any applicable
   carrying cost rate is the same for all
   maturities, then this is a reasonable assumption.
   This is because the primary cash-flow difference
   between comparable futures and forwards is the
   marking to market for futures.
3. As long as the same rate applies, any
   intermediate cash flows get invested at the same
   rate so there is no advantage of one over the
   other. However, if the futures price is
   positively (negatively) correlated with interest
   rates, then we will tend to reinvest gains at
   higher (lower) rates of interest and futures will
   have an advantage (disadvantage) over forwards.
   Futures prices will therefore be higher (lower)
   than forward prices.
4. Other reasons why comparable futures and forwards
   may have different prices are tax, margin,
   liquidity or transactions costs differences.
   These are ignored here.
5. As long as the maturity of the contract is short,
   even changing interest rates will have little
   effect on futures versus forward prices.

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Example Calculating returns on futures
-speculator
ASSUME - It is January - July wheat futures
sell for 4.84/bushel - Each contract covers
3000 bushels - Margin rate is 15 - Trading
commission is 30/roundtrip Buy 5
contracts Figure your investment 5 x 3000 x
4.84 x .15 (30 x 5) 11,040 A. Sell
your futures in April when price is 4.96 /
bushel .149/4 months or 44.8
annual B. Sell your futures in March when price
is 4.75/bushel -.136/3 months or
-54.3 annual
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Hedging Locks in Profit - Eliminates Price Risk

• SIMPLE HEDGING EXAMPLE - Mortgages
• A local bank makes commitments with customers
  for 5 million of mortgage loans today at a fixed
  rate of interest. The mortgages are not actually
  paid out until real estate closings in two
  months. The banks funding costs are 4 m.
• At the same time, an insurance company plans to
  buy 5 million in mortgage-backed securities in
  two months to support 6 million in insurance
  premiums it will receive.
• Local Bank - short hedger - hedges risk by
  selling futures.
• Insurance Company - long hedger - hedges by
  buying mortgage futures.
• Timing Local Bank Insurance Co
• now sell futures 5 buy futures -5
• 2 months later cost -4 gets premiums 6
• Net Profit 1 1

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Price Changes Have No Net Impact
1. Suppose mortgage rates rise so the value of
the 5 million mortgages is 3 million in two
months. What happens? Local Bank
Insurance Co. Makes mortgages 3 Buys
mortgages -3 Buy back futures -3
Sell back futures 3 NET 0
0 What are the respective gains and losses for
the Bank and the Insurance Co.? Local
Bank Insurance Co. Loss on mortgages -2
Gain on mortgages 2 Gain on futures 2
Loss on futures -2 NET 0
0 2. Suppose mortgage rates fall so the
value of the 5 million mortgages is 6 million
in two months. What happens? Local
Bank Insurance Co. Makes mortgages 6 Buys
mortgages -6 Buy back futures -6 Sell
back futures 6 NET 0 0 What are the
respective gains and losses for the Bank and the
Insurance Co.? Local Bank Insurance Co. Gain
on mortgages 1 Loss on mortgages -1 Loss
on futures -1 Gain on futures
1 NET 0 0
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Information in Futures Prices
Problem Suppose your company delivers oil to
customers at a fixed price of 1 per gallon. You
have an inventory of 1 million gallons and
storage capacity for 2 million gallons. Your
customers will be using 0.5 million gallons per
month over the next four months. It is January 1
and you observe the following set of prices for
spot oil and oil futures. Spot 0.90 per
gallon February 1.00 March 1.11 April 1.2
3 What is your strategy for purchasing the oil
you will need? If there are many firms in your
situation, how might spot and futures prices
change in the near-term? Problem Assume
everything above but you observe a new set of
prices for spot oil and oil futures. Spot 1.23
per gallon February 1.11 March 1.00 April
0.90 What is your strategy for purchasing the
oil you will need? If there are many firms in
your situation, how might spot and futures prices
change in the near-term?
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Problem Assume everything above but you observe
a new set of prices for spot oil and oil
futures. Spot 1.40 per gallon February 1.30
March 1.20 April 1.10 What is your strategy
for purchasing the oil you will need? If there
are many firms in your situation, how might spot
and futures prices change in the near-term?
(something like this happened in New England in
January 2000.) Redo each problem and assume that
you have 2 million gallons in inventory. Note
Futures prices signal information to market
participants and different price patterns can
induce different behaviors from participants.
Behavior also differs depending on inventory
levels.
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Pricing Futures

• Assume fixed supply and demand and no carrying
  costs (including zero interest rate and zero
  storage cost).
• QUESTION What should be the relation between
  spot
• and futures price?
• Ft,T St
• t a point in time, say, now
• T future date beyond time t
• Ft,T futures price covering time t to time
  T
• St spot price at time t
• This must occur or else arbitrage is possible
  because with fixed supply and demand, spot price
  will be the same in each future period.
• QUESTION If Ft,T gt St then what can you do to
  earn a
• risk free profit?
• Buy spot and store it. Short futures and deliver
  in the future to earn profit equal to (Ft,T -
  St).

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2. Relax assumption of zero carry costs A.
INTEREST CARRY COSTS Suppose interest rates are
positive, then interest payments make holding a
commodity (which pays no interest) less
attractive. It makes futures contracts more
attractive. This is like options where the
option price depends upon interest rates because
the value of competing positions depend on
interest rates. Thus with continuous
interest Ft,T Ster(T-t) B. For any
other carry costs we simply add a new term to the
exponent. For example, it costs something to
store the commodity such as grain silos,
insurance, etc. Assuming carry costs in
percent C and accrue continuously, then we
have the following Ft,T Ste(rc)(T-t)
(If the present value of storage cost is a
fixed C per unit of the commodity then Ft,T
(St C)er(T-t) ). C. There can also be a
benefit to holding an asset or commodity called a
convenience yield (Y) (e.g., dividend, coupon
payment, stock-out costs avoided) so that Ft,T
Ste(rc-Y)(T-t)
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3. Interest rates for the futures contract life
are easy to observe but some carry costs and
convenience yield are difficult to estimate and
often vary over time. For example, if there is
uncertainty about whether war will break out in
the Middle East, the futures market may price in
a large convenience yield for oil futures.
However, if some believe that demand will fall if
OPEC cuts off supply, then U.S. oil buyers may
not believe that there should be much of a
convenience yield. 4. Synthetic futures assume
that there are no carry costs except interest
then we can show how to price futures using a
replicating portfolio composed of options.
Suppose we buy a call on a commodity with an
exercise price of Ft,T (the current futures
price) priced at Ct and sell a put with the same
exercise and maturity (T) priced at Pt. Assume
that the spot price of the commodity is St. This
portfolio has the same payoff at maturity as a
futures contract if the spot price rises above
(falls below) the exercise price, then we gain
(lose) dollar for dollar. Then from Put-Call
parity we must have Ct Pt St - Ft,Te-r(T-t)
We know that the the futures contract price is
set such that there is no initial investment
(ignoring the margin), therefore, Ct Pt St -
Ft,Te-r(T-t) 0 gt Ft,T Ster(T-t) which is
the correct pricing formula.
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RELAX THE FIXED SUPPLY/DEMAND ASSUMPTION

• 4. Now the expected future spot price, which
  depends on
• expected future supply and demand, is important
• If the expected future spot is lower than the
  present spot price and there are costs of carry
  then the futures will be priced according to the
  expected future spot price.
• If the expected future spot is higher than the
  present spot price then the futures will be
  priced according to the present spot plus carry
  costs.
• One can always carry the commodity from the
  present into the future but you can't pull it
  back from the future.
• The general pricing relationship is then
• Ft,T MinEt(ST), Ste(rc-Y)(T-t)
• QUESTION What are examples of each alternative
• pricing method?
• Perishables - Strawberries, milk

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QUESTION How should zero coupon T-bond futures
be priced? QUESTION How should coupon T-bond
futures be priced? QUESTION How should SP
500 futures be priced? NOTE The futures price
does not include the expected return of the SP
over the contract period. This is because, if you
buy a contract you buy the systematic risk of the
SP and should be rewarded, that is Ft,T lt
Et(ST). - the seller is selling the risk and
the buyer buys it, so the buyer must be
compensated by paying a low Ft,T now and
expecting to receive a higher Et(ST) later.
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The SP 500 futures price is not set at Et(ST)
because Ste(rc-Y)(T-t) lt Et(ST) ACCORDING TO
THE FUTURES PRICING MECHANISM, WE GO WITH THE
LOWER PRICE. If the market expects an extra
return in the SP beyond its normal expected
return required for its risk then both St and
Ft,T will move up to discount the extra no-risk
return and thus the relationship Ft,T
Ste(rc-Y)(T-t) still holds.
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Another way to see that the futures price for
risky assets such as the SP 500 will be below
the expected future spot price is to assume that
the asset price grows at the assets expected
return k, which exceeds the risk-free rate r.
Assume there are no carry costs or dividends. A
speculator who buys the futures contract on the
asset and invests the present value of the
futures price in the risk-free asset has the
following cash flows Time t -Fe-r(T-t) put
money in risk-free investment now which will
be used to pay for the asset delivered on the
futures contract. Time T ST pay future spot
price on futures contract when futures mature
at time T. The expected net present value of this
investment must be zero given that both
investments are discounted at the appropriate
rates -Fe-r(T-t) ESTe-k(T-t) 0 Or F
ESTe(r-k)(T-t) Clearly, for any risky asset,
k gt r so that the futures price is always less
than the expected future spot price EST. Only
if the asset is risk free will the futures price
equal the expected future spot price.
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Time Series Behavior of Futures Prices
Samuelson (1965) showed that futures prices will
fluctuate randomly over time and that the
variance of futures prices may not be constant
over time. He assumes that St1 aSt et , a
lt 1 This simple autoregressive model says that
the spot price of the asset is expected to
decline over time. Nevertheless, the variance of
the expected spot price increases over time
because the error terms can accumulate over time
and leave the realized spot price far from its
expected value. Even though the spot price
changes in a known way, the futures price
(assumed to be the expected spot price at
delivery) is not expected to change. It already
reflects the expected spot changes over time
defined by the model above. At each point in
time, we know what St is and with the model above
we know what the expected future spot price is
(say for 2 periods ahead we have a2ESt). Past
errors feed in to determine St but the past
pattern of spot prices is irrelevant, only the
present spot price is relevant. Futures price
depends on it according to the model. The
variance of the futures price usually increases
as a contract approaches maturity. This is
because a futures contract for delivery far in
the future will have a price very close to that
determined from the model above. Even if we get a
large positive (negative) error this period that
makes St1 much larger (smaller) than expected,
it will likely be offset by opposite sign errors
over the numerous future periods. Also, if a is
small, then future spot price is likely to be
small far in the future no matter what the error
is in any near-term period.
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The error wont change our expectation for prices
far in the future because it is likely that the
error will be offset by other errors of the
opposite sign over time. However, if we get a
large error today, the near-term contracts have
little time left for other errors to offset a
recent large error. (this is similar to
Carmelos result that the discount rate for a
cash flow to be receive far in the future should
be about the risk-free rate until enough time
passes so that it starts to take on more
potential variance and risk) This example is one
way to show that there is not necessarily a
relationship between todays spot price and
todays futures price. Todays spot price depends
on what is happening today and todays futures
price depends mostly on what is expected to
happen well into the future at the delivery
date. Only if there is a link between the present
and the future will we see a link between spot
and futures prices. For example, oil prices may
be low today but if we know that war is likely to
break out in the Middle East in 6 months, then
the six month futures contract price will be
high. If people prepare for this possibility by
hoarding oil now, the present spot price will
also be high. If they dont hoard, then spot
prices may stay low. See oil or gas prices for
short and long contracts at cmegroup.com or
futures.quote.com.
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EXAMPLE TESTING SP FUTURES FOR ARBITRAGE
OPPORTUNITIES
Spot SP Nov. 6, 2001 1118.86 Futures
Dec. 2001 1121.00 Mar.
2002 1122.40 SP dividend yield 1 2 month
Tbill rate 1.7 5 month Tbill rate
1.8 Prompt date is the Thurs. before the third
Friday of Month Theoretical Price
Actual - Theoretical FN,D 1118.86e(.017 -
.01)(44/365) 1119.80
1121 1119.80 1.20 FN,M 1118.86e(.018 -
.01)(128/365) 1122.00 1122.4
1122 0.4 There are only slight differences
between actual and theoretical futures prices. It
would probably not pay to try to arbitrage since
trading commissions may exceed profits. Assuming
no cost to trade then for the December
contract Arbitrage opportunity - sell futures /
buy spot Sell futures now/ make delivery in
Dec. 1121 Buy spot now
-1118.86 Forgo interest on funds
(1118.86e.017(43/365) 1) -2.24
Receive dividends (1118.86e( .01)(44/365)
1) 1.30 Guaranteed profit no matter what
SP does 1.20
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Hedging With Futures

• When a futures contract on the exact commodity is
  available then hedge ratio is 1 to 1.
• 1 million bushels of 5 wheat -gt 1 million
  bushels in futures
• When a futures contract on a similar or related
  commodity exists then calculate a hedge ratio.
• For an effective hedge we want the change in the
  value of the spot commodity to be equal to minus
  the change in the value of the futures. The
  amount of futures needed per unit of spot is
• h hedge ratio - ?Spot price / ?Futures price
• - this means if the spot and futures price move
  together (opposite), sell (buy) futures to hedge.

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For example, suppose you hold IBM bonds but only
Treasury bond futures are available. You can
hedge your IBM position by knowing the change in
the price of your bond when the Tbond price
changes. Your bonds price change is (1) Py
Price of your bond Dury Duration of your
bond Yoy old yield of your bond Yny new
yield of your bond The Treasury bonds price
change is (2) PT price of treasury
bond DurT Duration of Treasury bond YoT Old
yield on Treasury bond YnT New yield on
Treasury bond
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To get the hedge ratio divide (1) by
(2) Where h the units (dollars)
of futures to be sold per unit (dollar) of spot.
This is reliable in most cases less reliable if
durations change much with rate changes. It
assumes a parallel shift in yield curve. NOTE
This hedges only interest rate risk - default
risk is ignored.
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EXAMPLECOMPLEX HEDGING - SOUTHEAST CORP

• 1. Assume that
• On Jan 6, 2001 Southeast authorized 60 million
  of 25 year bonds to fund a building project which
  would be needed in August 2001.
• Bonds are Aa rated and have Yield of 12.88 if
  issued today.
• The bonds have a duration of 7.8
• A regression of Aa yield changes on Tbond yield
  changes has a slope of 1.123
• The Tbond futures of September 2001 had a price
  of 69 - 08 or 69.25
• The futures contract price is 69,250 on a
  100,000 face value 8 contract
• The cheapest to deliver bond for the September
  contract has an 11.80 Yield.
• It also has a duration of 8.5 years

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FINDING THE NUMBER OF FUTURES CONTRACTS NEEDED TO
HEDGE

• FIND THE HEDGE RATIO
• h 7.8/8.5 x 1.123/1 x 1.118/1.1288 1.021
• FIND DOLLAR AMOUNT OF FUTURES NEEDED TO HEDGE
• F 60,000,000 1.021 61,260,000
• FIND NUMBER OF FUTURES CONTRACTS NEED
• NF 61,260,000/69,250 885
• Question How would we hedge a stock portfolio?
• You need the betas of the spot portfolio and the
  futures
• portfolio.
• h -Beta(Spot) / Beta(Futures)

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International Finance and Foreign Exchange Futures
SIMPLE DEFINITION Buying one currency with
another INTEREST RATE PARITY - implies that all
countries have the same interest rate after one
adjusts up or down for the change in the
country's currency value. INTEREST RATES ARE
THE PRICE OF MONEY - Thus the futures price of,
say, the dollar in terms of the Euro, will depend
on their relative prices, i.e., the respective
interest rates. Ft,T the future price of one
unit of foreign currency in terms of the
domestic currency e.g, 2/1. St the spot
price of one unit of foreign currency in terms
of the domestic currency. Rd,T-t the domestic
interest rate covering the contract
period. Rf,T-t the foreign interest rate
covering the contract period. Ft,T Ste(Rd -
Rf)(T-t)
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Like SP futures adjusted for dividends, here we
adjust for the rate earned on the foreign
currency. - if Rd Rf gt Ft St - if Rd gt
Rf gt Ft gt St - if Rd lt Rf gt Ft lt St You
can always exchange dollars for Euros and get
Euro interest rates so arbitrage forces interest
rates between countries to be the same adjusted
for expected currency depreciation or
appreciation. or, Ft,T / St e(Rd - Rf)(T-t)
Suppose Rd increases and Rf stays constant this
implies that Ft,T/St increases so St decreases or
Ft,T increases or both. WHAT OFTEN HAPPENS IS St
INCREASES BUT Ft,T INCREASES EVEN
MORE. QUESTION Why? ANSWER Because the spot
rate now will be a function of the expected
future spot rate - pure expectation
hypothesis investors hold currencies that they
expect to appreciate which increases the demand
for them now. Also, if a country increases its
interest rate (through the central bank),
investors might expect more increases and bid up
currency futures.
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EXAMPLE OF INTEREST RATE PARITY AND EXCHANGE RATES
ASSUME Spot rate of British pound is 1.70 per
pound The annual pound interest rate is RL
.11 The annual dollar interest rate is
R .13 QUESTION What should be the Futures
price of pounds to be delivered in one
year? F 1.70 x e(.13 - .11)(1) 1.734
dollars per pound
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Overall International

• Interest rate parity similar to PPP
  purchasing power parity
• International Cost of Capital Tom OBrien
• Lots of institutional details differences in
  accounting, taxes, trade zones, tariffs, transfer
  pricing, political risks, monetary and fiscal
  policies, letter of credit, etc.
• Hedging do shareholders expect a hedged cash
  stream or an unhedged cash stream?
• Revenues only foreign sales
• Costs only foreign production
• Profits Revenues Costs gt both foreign
• Assets only foreign plant
• Liabilities only foreign financing
• Equity Assets Liabilities gt both foreign

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Steps in the 1997-1998Asian Financial Crisis

• Thailand experiences major financial collapse
• 2. Russia defaults on government debt
• 3. World-wide rush to buy U.S. Treasury bonds
  (caused ITCM collapse Fed - banks - bail out).
• 4. Dollar appreciates intermediate mechanism
• 5. Real U.S. export (import) prices increase
  (decrease)
• 6. U.S. exports (imports) fall (rise)
• 7. U.S. interest rates fall intermediate
  mechanism
• 8. U.S. consumption (savings) rises (falls),
  investment rises

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Graphing the relevant data for Asian Financial
Crisis

• Go to economagic.com, click on Federal Reserve,
  St. Louis
• 2. Click U.S. Balance of Payments Data
• 3. Click Balance on Current Account (this is
  quarterly)
• 4. Click Gif Chart or PDF Chart (see recent and
  to 1960)
• 5. Click Foreign Assets in the United States, Net
  Capital Inflows (US Assets Abroad, Net Outflows)
• 6. Click U.S. Interest Rate Data. Then click on
  30-Year Treasury Constant Maturity or another
  maturity
• 7. Click Exchange Rate Data. Then click
  Trade-Weighted Exchange Index Broad

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8. For Export and Import Prices go to
www.bls.gov 9. Click Databases Tables tab 10.
Go to Prices International and click on Top
Picks 11. Select Imports All Comodities 12.
Select Exports All Comodities 13. Click
Retrieve Data. 14. Click Include Graphs and then
click Go. 15. Click More Formatting
Options. 16. Select 12-Month Percent Change and
Retrieve Data. 17. You can see that import
prices fell faster than export prices around
1997-1998.