Finance
School of Management
Chapter 14: Forward &
Futures Prices
Objective
• How to price forward and futures
• Storage of commodities
• Cost of carry
• Understanding financial futures
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School of Management
Finance
Chapter 14: Contents







Distinction Between Forward &
Futures Contracts
The Economic Function of
Futures Markets
The Role of Speculators
Relationship Between
Commodity Spot & Futures
Prices
Extracting Information from
Commodity Futures Prices
Spot-Futures Price Parity for
Gold
Financial Futures






The “Implied” Risk-Free Rate
The Forward Price is not a
Forecast of the Spot Price
Forward-Spot Parity with Cash
Payouts
“Implied” Dividends
The Foreign Exchange Parity
Relation
The Role of Expectations in
Determining Exchange Rates
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Features of Forward Contracts

Two parties agree to exchange some item on a specified
future date at a delivery price specified now.

The forward price is defined as the delivery price
which makes the current market value of the contract
zero.

No money is paid in the present by either party to the
other.

The face value of the contract is the quantity of the
item specified in the contract times the forward price.
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Features of Forward Contracts

The party who agrees to buy the specified item is said
to take a long position, and the party who agrees to
sell the item is said to take a short position.

“Customization”, difficulty of “closing out” positions,
low liquidity

The risk of contract default, credit risk
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Characteristics of Futures

Futures are
– standard contracts
– immune from the credit worthiness of buyer
and seller because
 exchange stands between traders
 contracts marked to market daily
 margin requirements (enough collateral)
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Terms

Futures List
Monday Aug.5,2007
WHEAT(CBT)5,000bu; cents per bu
Open
High
292
2941/2 289



Low
Settle
2943/4
Change
Lifetime High
-71/4
326
Lifetime Low Open Interest
258
16.168
Mark-to-market
Margin requirement
Margin call
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An Illustration

You place an order
to take a long
position in a
September wheat
futures contract on
August 4, 1991.

The broker
requires you to
deposit an initial
margin of $1,500 in
your account.

On August 5, the futures price closes 71/4 cents per
bushel lower.
– You have lost 71/4 cents*5,000 bushels = $362.50
that day.
– Marking to Market: the broker takes that amount
out of your account and transfers it to the future
exchange, which transfers it to one of the parties
who was on the short side of the contract.
– If you do not have enough money in your account
to meet the margin requirement (variation /
maintenance margin), you’ll receive a margin call
from the broker asking you to add money.
– If you do not respond immediately, then the
broker liquidates your position at the prevailing
market price.
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Spot-Futures Price Parity for Gold

There are two ways to invest in gold.
– buy an ounce of gold at S0, store it for a year at a
storage cost of $hS0, and sell it for S1.
– invest S0 in a 1-year T-bill with return rf , and purchase a
1-ounce of gold forward, F, for delivery in 1-year.
S1  S 0
S F
 h  rgold  rgold( syn)  1
 rf
S0
S0
F  (1  r f  h)S0
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Arbitrage Opportunities of Forwards:
An Illustration

The spot price of gold is $300, the storage costs is 2%
per year, and the risk-free rate is 8%.
– If the forward price is $340 (too high)
Arbitrage Position
Sell a forward contract
Borrow $300
Buy an ounce of gold
Pay storage costs
Net cash flows
Immediate Cash
Flow
Cash Flow 1 Year
from Now
0
$300
-$300
$340-S1
-$324
S1
-$6
$340-$330=$10
0
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– If the forward price is $320 (too low)
Arbitrage Position
Sell short an ounce of gold
Buy a forward contract
Invest $300 in 1-year pure
discount bond
Receive storage costs
Net cash flows
Immediate Cash
Flow
$300
0
Cash Flow 1 Year
from Now
-$300
$324
0
$6
$330-$320=$10
0-S1
S1 -$320
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Spot-Futures Price Parity for Gold

A contract with life T:
F  1  rf  h  S 0
T
– This is not a causal relationship, but the forward and
current spot are jointly determined in the market.
– If we know one, then the law of one price determines
that we know the other.
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Implied Cost of Carry

Implied cost of carry = F − S0 = (rf + h) S0

The implied cost of storing the gold (per $spot) is
h = (F − S0)/S0 − rf

Suppose F = $330, S0 = $300, and rf = 8%, then
– Implied cost of carry = $330 − 300 = $30
– Implied storage cost = (330 − 300)/300 − 8% = 2%
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Financial Futures
Financial futures contracts are usually settled
in cash.
 With no storage cost, the relationship between
the forward and the spot is


F  S 1 rf
T
– Any deviation from this will result in an arbitrage
opportunity.
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Replication of Non-Dividend-Paying Stock
Using a pure Discount Bond and a Stock
Forward Contract
Position
Buy a share of stock
Immediate Cash Flow
Cash Flow 1 Year
from Now
–S
S1
Replicating Portfolio (Synthetic Stock)
Go long a forward contract
0
on stock
Buy a pure discount bond
– F/(1+r f )
with face value of F
Total replicating portfolio
– F/(1+r f )
S1 – F
F
S1
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Arbitrage in Stock Futures

The spot price of a stock is $100, and the risk-free rate
is 8%.

The forward price is $109.
Arbitrage Position
Sell a forward contract
Borrow $100
Buy a share of stock
Net cash flows
Immediate Cash
Flow
0
$100.00
-$100.00
0
Cash Flow 1 Year
from Now
$109-S1
-$108.00
S1
$1
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The “Implied” Risk-Free Rate

Rearranging the formula, the implied interest
rate on a forward given the spot is
1
T
F
F  S0
r     1; if T  1, r 
S0
 S0 
– This is reminiscent of the formula for the interest
rate on a discount bond.
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Replication of a Pure Discount Bond
Using a Stock and a Forward Contract
Position
Immediate Cash
Flow
Cash Flow 1 Year
from Now
Buy a T-bill with face value
– F/(1+r f )
of F
Replicating Portfolio (Synthetic T-Bill)
Buy a share of stock
–S
Go short a forward contract
0
Total replicating portfolio
–S
F
S1
F – S1
F
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The Forward Price is not a Forecast
of the Spot Price

Following the diagrams in Chapter 13 we might
suppose that the expected price of a stock is
S1  S 0 (1  r f  risk premium)  S 0 (1  r f )  F

If this is indeed correct, then the forward price is not
an indicator of the expected spot price at the maturity
of the forward.
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Forward-Spot Parity with Cash Payouts

The S − F relationship becomes
DF
S
 F  S  rf S  D
1 rf
– Note: (forward price > the spot price) if (D < rf S)
– Because D is not known with certainty, this is a
quasi-arbitrage situation.
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Replication of Dividend-Paying Stock
Using a Pure Discount Bond and a Stock
Forward Contract
Immediate Cash
Flow
Buy a share of stock
-S
Replicating Portfolio (Synthetic Stock)
Go long a forward contract
0
on stock
Buy a pure discount bond
- (D + F)/(1+r f )
with face value of D + F
Position
Cash Flow 1 Year
from Now
D + S1
S1 – F
D+F
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“Implied” Dividends

From the last slide, we may obtain the implied
dividend


D  1 rf S  F
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Exchange Rate Example
Time
Japan
15000 ¥
(Borrowed)
U.K.
150 ¥/£
3% ¥/¥ (direct)
3% ¥/£/£/¥
15450 ¥
15450 ¥
(Repaid)
£100
(Invested)
9%£/£
Forward ¥/£
£109
(Matures)
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The Foreign Exchange Parity Relation

We used the diagram to show that
$ Denominated Forward on Yen
1  r$ 
t

$ Denominated Spot for Yen
1  rY t
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