# Forward and Future Contracts

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```					  Lecture 7. Pricing Forward and Future Contracts & Interest
Rate Contracts

Relating forward and future prices to the spot price
&
Hedging exposure to interest rate movements

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Pricing Futures Contracts vs. Pricing Forward Contracts

Forward contract pricing: F0 = S0erT . If interest rates vary then forward and future prices may
no longer be the same.

However, if the risk-free interest rate is constant and the same for all maturities, then the forward
price for a contract is the same as the futures price. We sketch a proof.

•   Suppose futures contract last n days
• Fi is the futures price at the end of day i with daily settlement
• Deﬁne δ as the risk-free rate per day.
• Strategy:
– Take a long futures position of eδ at the start
– Increase long position to ekδ at end of day k
• At end of day i the position is (Fi − F i − 1)eδi. Assume proﬁt is compounded at risk-free
rate until end of day n
iδ (n−i)δ                     nδ
(Fi − Fi−1) e e               = (Fi − Fi−1) e
Pn                   nδ
• Value at end of n days is   i=1 (Fi − Fi−1 ) e    = (Fn − F0) enδ .
• Fn = ST , the terminal spot price, so value of the strategy is (ST − F0) enδ .

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Pricing Futures Contracts vs. Pricing Forward Contracts

• An investment of F0 in a risk-free bond, combined with above strategy yields
nδ                   nδ            nδ
F0e        + (ST − F0) e        = ST e

• On the other hand, a forward contract with price G0 at day 0 yields a return of ST enδ .
• In absence of arbitrage opportunities implies F0 = G0

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Futures on Commodities

We consider pricing futures contracts on commodities that are primarily
investment assets, such as precious metals.

• Gold owners can earn income from leasing the gold.

• The interest charged from leasing gold is the gold lease rate

• Also holds for silver.

In the absence of storage and income, the forward price of the commodity that is
an investment asset is
F0 = S0erT
We treat storage costs as negative income, then set U to be the present value of
all storage costs, net of income, during the life of a forward contract, then

F0 = (S0 + U ) erT

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Futures on Commodities, cont.

Example: Consider a 1 year futures contract on an investment asset that provides
no income. It costs \$2 per unit to store the asset, with the payment being made
at the end of the year. Assume that the spot price \$450 per unit and the risk-free
rate is 7% per annum for all maturities. This corresponds to r = 0.07, S0 = 450,
T = 1, and
U = 2e−0.07×1 = 1.865
The futures price, F0 is given by

F0 = (450 + 1.865) e0.07×1 = \$484.63

If the actual futures price is greater than 484.63, an arbitrageur can buy the asset
and short 1-year futures contracts to lock in a proﬁt. If the actual futures price is
less than 484.63, an investor who owns the asset can improve the return by selling
the asset and buying futures contracts.

If storage costs are proportional to the price of the commodity then it can be
treated as negative yield:
F0 = S0e(r+u)T
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Consumption Commodities

Commodities that are consumption assets (as opposed to investment assets) usually provide no
income, but can be subject to signiﬁcant storage costs. We consider if our pricing model holds:
rT
F0 > (S0 + U ) e

Arbitrageur should follow the following strategy:

• Borrow an amount S0 + U at the risk-free rate and use it to purchase one unit of the
commodity and to pay for storage.
• Short a forward contract on one unit of the commodity.

This would ideally lead to a proﬁt of F0 − (S0 + U )erT at time T . However, there is a tendency
for S0 to increase and F0 to decrease until F0 > (S0 + U ) erT no longer holds. Therefore,
F0 > (S0 + U ) erT cannot hold for long periods of time.

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Consumption Commodities

On the other hand, consider
rT
F0 < (S0 + U ) e
Then an investor should do the following:

• Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interest rate.
• Take a long position in a forward contract.

The result is a riskless rate of (S0 + U )erT − F0 relative to the position they would’ve been had
they held the commodity. Therefore, we seem to get F0 = (S0 + U ) erT for the pricing of the
commodity.

However, companies who keep consumption commodities in storage, do so because of its
consumption value, not because of investment value. Such owners are reluctant to sell the
commodity and buy forward contracts, because forward contracts cannot be consumed. There is
nothing to stop F0 < (S0 + U ) erT from holding.

All we can say is
rT
F0 ≤ (S0 + U ) e
If storage is proportional to the spot price then
(r+u)T
F0 ≤ S0e

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Convenience Yields

Users of the consumption asset place diﬀerent value to the commodity than investors.

• Users of corn futures keep production running
• Beneﬁts from holding the physical asset are referred to as the convenience yield provided by the
community
• If the cost of storage is known and has present value U then the convenience yield y is deﬁned
as
yT                rT
F0e = (S0 + U ) e
• If cost of storage is a proportional u to the spot price, then y is
(r+u−y)T
F0 = S0e

Convenience yield measures the extent to which the left-hand side
is less than the right-hand side in F0 ≤ (S0 + U ) erT or F0 ≤ S0e(r+u)T .

Reﬂects the market’s expectations concerning the future availability of the commodity.

• Greater the possibility of shortages, greater the convenience yield
• Larger the inventories, the smaller the convenience yield

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Cost of Carry

Cost of carry is storage costs and interest that is paid to ﬁnance the asset less the income earned
on the asset.

•   Nondividend-paying stocks, the cost of carry is r (no storage & no income)
•   Stock index, the cost of carry is r − q (income earned at rate q )
•   Currency futures, the cost of carry is r − rf (diﬀerence in the foreign risk-free rate)
•   Commodity with income q and storage costs at rate u, the cost of carry is r − q + u

Cost of carry is
cT
F0 = Soe
for investment assets and
(c−y)T
F0 = S0e
for consumption assets with y the convenience yield.

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Futures prices and Expected futures prices

• The market’s average opinion about what the spot price of an asset will be at a certain time in
the future is the expected spot price
• Speculators tend to hold long positions, since require greater compensation for the greater risk.
Only trade if they expect to make money.
• Hedgers tend to hold short positions and more willing to take losses, since their primary interest
is reducing risk.
• Together this implies futures price of an asset will be below the expected spot price - Keynes &
Hicks.

We see that

• Relationship between futures prices and expected short is based on the relationship between risk
and expected return
• More later on

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Summary

• Futures and forward prices is convenient to divide futures contracts into two categories
• Pricing occurs via arbitrage situation
• Three cases, assume
– Assets with no income: F0 = S0erT and
Value of long foward contract with delivery price K is S0 − Ke−rT
– Assets with known income: F0 = (S0 − I)erT and
Value of long foward contract with delivery price K is S0 − I − Ke−rT
– Assets with known yield: F0 = S0e(r−q)T and
Value of long foward contract with delivery price K is S0e−qT − Ke−rT
• Futures prices for stock indices, currencies, gold, and silver can be priced in the same way.
• Pricing consumption asset futures need to take into account the beneﬁts of owning the asset
that are not obtained from owning the futures contract. This is measured by the convenience
yield.
• Can obtain only upper bounds for futures price of consumption assets using arbitrage
arguments.
• Cost of carry is the storage cost plus ﬁnancing minus the income. For investment assets futures
price is greater than the spot price by the amount reﬂecting the cost of carry. For consumption
assets futures price is greater than the spot price by an amount reﬂecting the cost of carry net
of the convenience yield.
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Interest Rate Futures

• Use Treasury bonds and Eurodollar futures contracts to hedge exposure to
interest rate movements.

• Interest rate futures allow companies to hedge themselves from risks associated
to changes in the risk-free rates.

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Day Count Conventions

Deﬁnes the way in which interest is accrued over time. Day count is expressed in
the form X/Y when computing interest between two dates.

Number of days between dates
× Interest earned in reference period
Number of days in reference period

Day count conventions provide for a way to determine the value of partial coupon
payments.

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Day Counts, cont.

• US Treasury Bonds - Day counted as

actual/actual reference period

The interest earned between dates is the ratio of the number of days to the true length of the
period of time.

Example: Suppose a bond principal is \$100, coupon payment dates are March 1 and
September 1, the coupon rate is 8%. What is the interest earned between March 1 and July 3?
The length of the time between March 1 and September 1 is 184 days, in which \$4 of interest
is earned. during the period. There are 124 days between March 1 and July 3. Therefore,

124
× 4 = \$2.6957
184

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Day Counts, cont.

• US Corporate and Municipal Bonds - Day counted as

30/360

Here, we assume 30 days per month and 360 days per year when carrying out calculations.

Example: Consider the same example. Suppose a bond principle is \$100, coupon payment
dates are March 1 and September 1, the coupon rate is 8%. What is the interest earned
between March 1 and July 3 using the 30/360 rule?
We count the length of the time between March 1 and September 1 as 180 days, in which \$4 of
interest is earned. during the period. The total number of days counted between March 1 and
July 3 is
(4 × 30) + 2 = 122
so the interest earned is
122
× 4 = \$2.7111
180

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Day Counts, cont.

• US Money Market Instruments - Day counted as

actual/360

Used for money market instruments in the US. Here, we calculate the actual number of days
that have elapsed and divide by 360 days per year when carrying out calculations. Note -
interest earned on an entire year is 365/360 times the quoted rate.

Sometimes the money market rates are quoted using discount rates.

Example: Consider the same example. Suppose a bond principle is \$100, coupon payment
dates are March 1 and September 1, the coupon rate is 8%. What is the interest earned
between March 1 and July 3 using the actual/360 rule?
We count the length of the time between March 1 and September 1 as 180 days, in which \$4 of
interest is earned. during the period. The total number of days counted between March 1 and
July 3 is 124 so the interest earned is

124
× 8 = \$2.7555
360

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US Money Market Instruments, cont.

Consider a 91-day Treasure Bill with annualized rate of interest earned at 8% of face value.
(91-day T-Bill quoted as 8)
Suppose face value \$100 then interest earned over the life of the Treasury Bill is

91
\$100 × 0.08 ×       = \$2.0222
360

The true interest rate is
2.0222
= 2.064%
100 − 2.0222
over the 91-day period.

In general we have
360
P =        (100 − Y )
n
where P is the quoted price, Y is the cash price, n is the remaining life of the Treasury bill in days.

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Quotes for Treasury Bonds

• US Treasury bonds are quote in dollars and thirty-seconds of a dollar. The
quote price is for a bond with face value of \$100.

• Example: A quote of 92-07 indicates a quote price for a bond with face value
\$100 is \$92.21875 .

• Quoted price is referred to as the clean price

• Cash price is referred to as the dirty price

Cash price = Quoted price + Accrued interest since last coupon date

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Quotes for Treasury Bonds

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Quotes for T-Bonds

• Example: Suppose it is March 5, 2007, and the bond is 11% coupon bearing
bond maturing on July 10, 2010 with quoted price of 95-16 ≡ \$95.50.

• Coupons for T-bonds are paid semiannually, so most recent coupon was
January 10, 2007 and next coupon date is July 10, 2007.

• 54 days between January 10, 2007 and March 5, 2007.

• 181 days between January 1. 2007 and July 10, 2007.

• On a bond with face value \$100, coupon payments are \$5.50. The accrued
interest on March 5, 2007 is the share of the July 10 coupon accruing to the
bondholder on March 5, 2007
54
× \$5.5 = \$1.64
181

• Case price for the \$100 face value bond is

\$95.5 + \$1.64 = \$97.14
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Treasure Bond Futures

• Treasury bond futures prices are quoted

• Delivery can occur anytime during the delivery month

• Each contract is for delivery of \$100,000 face value of bonds.

• Treasury bond futures are traded on CBOT.

• Any bond with 15 years to maturity on the ﬁrst day of the delivery month and
is not callable within 15 years from that day can be delivered

• A callable bond contains provisions in which it can be bought back at a certain
price at certain times during its life

• Treasury Note Futures Contracts with a maturity between 6 1/2 and 10 years
can be delivered.

• 5 year Treasury Note Futures, any bond delivered has a life of about 4 or 5 years
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Conversion Factors

• Treasury bond futures contract allows the short position to choose to deliver any bond with
maturity of more than 15 years and is not callable within 15 years.
• When a bond is delivered, the conversion factor deﬁnes the price received by the party with the
short position.
• The quoted price applying to the delivery is conversion factor times most recent settlement
price. If we include accrued interest then the cash received for a \$100 face value of bond
delivered is
(Settlement price × Conversion factor) + Accrued Interest

Example: Consider a bond with settlement price of 90-00 with conversion factor for the bond
delivered of 1.3800. Assume the accrued interest at the time of delivery is \$3 per \$100 face value.
The cash received by the short investor is

(1.3800 × 90.00) + 3.00 = \$127.20

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Calculating Conversion Factors

• Conversion factor for a bond is equal to the quoted price the bond would have per dollar of
principal on the ﬁrst day of the delivery month on the assumption that the interest rate for all
maturities would equal 6% per annum with semiannual compounding
• The bond maturity and the times to the coupon payments are rounded down to the nearest 3
months for calculation purposes.
• After rounding, if the bond does not last for an exact number of 6-month periods (i.e. and
extra 3 month period), the ﬁrst coupon is assume to be paid after 3 months and accrued
interest is subtracted.

Example: Consider a 10% coupon bond with 20 years and 2 months to maturity.

• Coupon payments are assumed to be made at 6-month intervals until the end of the 20 years
when the principal is paid.
• Assume the face value is \$100
• Given the discount rate of 6% per annum with semiannual compounding, then the face value is
40
−i                  −40
X
5 (1.03)             + 100 (1.03)         = \$146.23
i=1

Dividing by the face value gives a conversion factor of 1.4623.
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Calculating Conversion Factors, Example 2

Example: Consider a 8% coupon bond with 18 years and 4 months to maturity.

• For the purpose of calculating the conversion factor, the bond is assumed to have exactly 18
years and 3 months to maturity.
• Discounting at 6% per annum compounded semiannually yields
36
X          4       100
4+                   i
+      36
= \$125.83
i=1
1.03     1.03
√
• Interest rate for a 3 month period is 1.03 − 1 = 0.014889.
• Discounting back to the present gives a bond value as

125
= \$123.99
1.014889

• Subtracting oﬀ the accrued interest of 2.0 yields \$121.99 . Conversion factor is 1.2199.

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Cheapest-to-Deliver Date

• Any time during the delivery month, there are many bonds that can be delivered in CBOT
Treasury bond futures contract.
• They can vary widely.
• Party with the short position can choose which of the available bonds is ”cheapest” to deliver.

Short party receives:

(Settlement price × Conversion factor) + Accrued interest

Cost of purchasing is:
Quoted bond price × Accrued interest
Cheapest-to-deliver bond is the one which

Quoted bond price − (Settlement price × Conversion factor)

is the smallest.

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Cheapest-to-Deliver Date

Consider a short portfolio of three bonds. Assume the most recent settlement price is 93-08.

Bond        Quoted bond price            Conversion Factor
1           99.50                        1.0382
2           143.50                       1.5188
3           119.75                       1.2615

The cost of delivery of the bonds are

Bond 1 :99.50 − (93.25 × 1.0382) = \$2.69
Bond 2 :143.50 − (93.25 × 1.5188) = \$1.87
Bond 3 :119.75 − (93.25 × 1.2615) = \$2.12

Cheapest-to-deliver is Bond 2.

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Cheapest-to-Deliver, cont.

Recall the smallest

Quoted bond price − (Settlement price × Conversion factor)

deﬁnes the cheapest-to-deliver bond. Consider a short portfolio of four bonds. Assume the most
recent settlement price is 101-12.
Bond        Quoted bond price             Conversion Factor
1           125-05                        1.2131
2           142-15                        1.3792
3           115-31                        1.1149
4           144-02                        1.4.026
The cost of delivery of the bonds are

Bond 1 :125.15625 − (101.375 × 1.2131) = \$2.178
Bond 2 :142.46875 − (101.375 × 1.3792) = \$2.652
Bond 3 :115.96875 − (101.375 × 1.1149) = \$2.946
Bond 4 :144.06250 − (101.375 × 1.4026) = \$1.874

Cheapest-to-deliver is Bond 4.
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Determining Futures Price

Assume that the cheapest-to-deliver and delivery dates are known, the T-Bond futures contract is
a futures contract on a security providing the holder with known income. Therefore,
rT
F0 = (S0 − I) e

where I is the present value of the coupons during the life of the futures contract, T is the time
until the futures contract matures, and r is the risk-free interest rate applicable to a time period of
length T .

Example:

• Consider a Treasury bond futures contract. It is known that the cheapest-to-deliver bond will
be a 12% coupon bond with a conversion factor of 1.4000.
• Suppose also that it is known that delivery will take place in 270 days.
• Coupons are payable semiannually on the bond
• The last coupon date was 60 days ago
• Next coupon date is 122 days from now and another in 305 days.
• The rate of interest is 10% per annum with continuously compounding

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Example, cont.

• The current quoted bond price is \$120.
• Cash price is obtained by adding the proportion of the next coupon payment that accrues to
60
the hold. The cash price becomes: 120 + 60+122 = 121.978
• A coupon of \$6 will be received after 122 = 0.3342 years. The present value becomes
365
−0.1×0.3342
6e             = 5.803
• Futures contract lasts for 270 = 0.7397 years.
365
• The cash futures price, if the contract were written on the 12% bond, is computed by
subtracting oﬀ the accrued interest

148
125.094 − 6 ×                 = 120.242
148 + 35

• From the deﬁnition of the conversion factor, 1.4000 standard bonds are considered equivalent
to each 12%. The quoted futures price should be

120.242
= 85.887
1.4000
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Example 2

We do another example of pricing. It is July 30, 2005. The cheapest-to-deliver bond in a
September 2005 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to
be made on September 30, 2005. Coupon payments on the bond are made on Feb. 4 and Aug. 4
each year. The term structure is ﬂat, and the rate of interest with semi-annual compounding is
12% per annum. The conversion factor for the bond is 1.5. The current quoted bond price is \$110.
What is the quoted futures price for the contract?

Recall F0 = (S0 − I) erT .

• There are 177 days between Feb. 4 and July 30 and 182 days between Feb. 4 and Aug. 4. The
cash price of the bond is, therefore;

177
110 +          × 6.5 = 116.32
182

• The rate of interest with continuous compounding is rC = 2 ln(1 + .12 ) = 0.1165 per year.
2
• A coupon of 6.5 will be received in 5 days = 001366 years time. The present value of the
coupon is
−0.01366×0.1165
6.5e                 = 6.490

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Example 2, cont.

• The futures contract lasts for 62 days = 0.1694 years. The cash futures price if the contract
were written on the 13% bond would be
0.1694×0.1165
(116.32 − 6.490) e                      = 112.02

• At deliver there are 57 days of accrued interest. The quoted futures price if the contract were
written on the 13% bond would be
57
112.02 − 6.5 ×     = 110.01
184

• Using the conversion factor we get the quoted price of

110.01
= 73.34
1.5

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Eurodollar Futures

• The most popular interest rate futures contract in the US is the 3-month Eurodollar futures
contract traded on CME.
• A eurodollar is a dollar deposited in a US or foreign bank outside the US.
• The Eurodollar interest rate is the rate of interest earned on Eurodollars deposited by one bank
with another bank.
• Essentially the same as the LIBOR (London Interbank Oﬀer Rate)
• 3-month Eurodollar futures contracts are contracts on the 3-month Eurodollar interest rate
• Allow for investors to lock in an interest rate on \$1 million for a future 3-month period.
• Delivery months of March, June, September, December for up to 10 years into the future

In other words, an investor in 2007 can use Eurodollar futures to lock in an interest rate for
3-month periods that are as far into the future as 2017.

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Eurodollar Futures, cont.

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Eurodollar Futures, cont.

• Consider a March 2005 contract.
• Settlement price is 97.63 and the contract ends on the third Wednesday of the delivery month
= March 16, 2005.
• The contract is marked to market in the usual way until this date.
• On March 16, 2005 the settlement price is set equal to 100 − R, where R is the actual
3-month Eurodollar interest rate on that day, expressed with quarterly compounding and an
actual/360 day count convention
• In particular if the 3-month Eurodollar interest rate on March 16, 2005, turned out to be 2%,
the ﬁnal settlement price would be 98.
• Final marking to market to reﬂect this settlement price and all contracts are declared closed

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Eurodollar Futures, cont.

Remarks:

• Contract designed so that each basis point (0.01%) results in a gain/loss of \$25 for each
long/short contract. One basis point up results in a gain of \$25 for the long and a loss of \$25
for the short. One basis point down results in loss of \$25 for the long and a gain of \$25 for the
short since
1
1, 000, 000 × 0.0001 × = 25
4
• Since the futures quote is 100 minus the futures interest rate, an investor who is long gains
when the interest rates fall and an investor who is short gains when interest rates rise.

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Eurodollar Futures, cont.

Example: On Feb. 4, 2004 an investor wants to lock in the interest rate that will be earned on \$5
million for 3 months starting on March 16, 2005.

• The investor goes long ﬁve March05 Eurodollar futures contract at 97.63
• On March 16, 2005 the 3=month LIBOR interest rate is 2%, so the ﬁnal settlement price
proves to be 98.00.
• The investor gains
5 × 25 × (9800 − 9763) = \$4625
on the long futures position.
• The interest earned on the \$5 million for 3 months at 2% is

5, 000, 000 × 0.25 × 0.02 = 25, 000

or \$25,000. The gain on the futures contract brings this up to \$29,625. This is the interest
that would have been earned if the interest rate had been 2.37%, i.e.
(5, 000, 000 × 0.25 × 0.0237 = 29, 625).
• Thus the interest rate is locked in at 2.37% = 100 - 97.63.

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Eurodollar, cont.

The exchange deﬁnes the contract price as

10, 000 × (100 − 0.25 × (100 − Q))

where Q is the quote. The settlement price of 97.63 for the March 2005 contract
(from our table) corresponds to a contract price of

10, 000 × (100 − 0.25 × (100 − 97.63)) = \$994, 075

In the above example, the ﬁnal contract price is

10, 000 × (100 − 0.25 × (100 − 98)) = \$995, 000

Diﬀerence between the initial and ﬁnal contract price is \$925, so the investor with
the long position in 5 contracts gains 5 × 925 = \$4, 625.

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Homework

Due Oct. 3, 5PM.

• 5.6, 5.9, 5.15

• Graded: 4.27, 5.24, 5.25, 5.27

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