# Determination of Forward and Futures Prices(1)

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```					  Determination of
Forward and Futures
Prices

Chapter 5
(all editions)
Consumption vs Investment
Assets
• Investment assets are assets held by
significant numbers of people purely for
investment purposes (Examples: gold,
silver, stocks, bonds)
• Consumption assets are assets held
primarily for consumption (Examples:
copper, oil, pork bellies)
Short Selling
• Short selling involves selling securities you do
not own
• Your broker borrows the securities from
another client and sells them in the market in
the usual way
• At some stage you must buy the securities
back so they can be replaced in the account of
the client
• You must pay dividends and other benefits to
the original owner of the securities
Notation

S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
Forward Price on Investment Asset
• For any investment asset that provides no
income and has no storage costs
F0 = S0erT
Example: Long forward contract to purchase
a non-dividend paying stock in three
months; current stock price is \$40, risk free
rate is 5%. Current forward price?
0.05(0.25)
F0 = 40e                = \$40.50
When an Investment Asset
Provides a Known Dollar Income
rT
F0 = (S0 – I )e
where I is the present value of the income
Example: Long forward contract to purchase a
coupon bearing bond in nine months which
provides \$40 coupon in 4 months; current price
is \$900 while the 4 month and 9 month risk free
rates are 3% and 4%, respectively. What is the
current forward price?
F0 = (900.00-40e-0.03*4/12)e0.04*9/12 = \$886.60
Arbitrage Opportunities
If F0 > (S0 – I )erT , F0 = \$910.00

Action now:
-Borrow                  \$900.00
• \$39.60 for 4 months at 3%
• \$860.40 for 9 months at 4%
-Sell forward for \$910.00

In 4 months:
-Receive \$40 income on asset to pay off the \$39.60e0.03*4/12 = \$40.00 first
loan with interest

In 9 months:
-Sell asset for \$910.00
-Use \$860.40e0.04*9/12 = \$886.60 to repay the second loan with interest

Profit realized: 910.00 – 886.60 = \$23.40
Arbitrage Opportunities
If F0 < (S0 – I )erT , F0 = \$870.00

Action now:
-Short asset to realize \$900.00
-Invest
• \$39.60 for 4 months at 3%
• \$860.40 for 9 months at 4%

In 4 months:
-Receive \$39.60e0.03*4/12 = \$40.00 interest on investment and pay income of
\$40 on asset

In 9 months:
-Receive \$860.40e0.04*9/12 = \$886.60 from investment

Profit realized: 886.60 – 870.00 = \$16.60
When an Investment Asset
Provides a Known Yield
(r–q )T
F0 = S0 e
where q is the average yield during the
life of the contract (expressed with
continuous compounding)
Value of a Forward Contract today
• Suppose that
-K is delivery price in a forward contract
-F0 is current forward price for a contract that
was negotiated some time ago
• The value of a long forward contract, ƒ, is
ƒ = (F0 – K) e–rT
• Example (pg 106)
• Similarly, the value of a short forward contract is
(K – F0)e–rT
• Similarly, one can determine the value of long
forward contracts with no income, known income
and know yield
Futures Prices of Stock Indices
• Can be viewed as an investment asset
paying a dividend yield
• The futures price and spot price
relationship is therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the
portfolio represented by the index
Example (pg 109)
Index Arbitrage
• When F0>S0e(r-q)T an arbitrageur buys
the stocks underlying the index and
sells futures
• When F0<S0e(r-q)T an arbitrageur buys
futures and sells (shorts) the stocks
underlying the index
Futures and Forwards on Currencies
• A foreign currency is similar to a security providing a
dividend yield
• The continuous dividend yield is the foreign risk-free
interest rate
• It follows that if rf is the foreign risk-free interest rate

Eg: 2-year interest rates in Australia and US are 5%
and 7%, respectively and the spot exchange rate is
0.6200 USD per AUD. The two year forward
exchange should be:
Why the Relation Must Be True
Arbitrage on Currency Forwards
Suppose 2-year forward exchange rate is 0.6300 USD per AUD
Action now:
•    AUD is cheaper; Borrow 1,000 AUD at 5% per annum for 2 years
and convert to 620 USD at spot exchange rate and invest the
USD at 7%
•    Enter into a forward contract to buy 1,105.17 AUD for 696.26
USD (1,105.17 x 0.6300)

In two years:
•    620 USD grows to 620e0.07*2 = 713.17 USD
•    The 1,105.17 AUD is exactly enough to repay principal and
interest on the 1,000 AUD borrowed (1000e0.05*2 = 1,105.17
AUD)
•    Need to buy 1,105.17 AUD under the forward contract; of the
713.17 USD, we use 696.26 USD to do so (696.26/0.6300)
•    Riskless profit of 713.17 – 696.26 = 16.91 USD
Arbitrage on Currency Forwards
Suppose 2-year forward exchange rate is 0.6600 USD per AUD
Action now:
•    USD is cheaper; Borrow 1,000 USD at 7% per annum for 2 years
and convert to 1,612.90 AUD at spot exchange rate and invest
the AUD at 5%
•    Enter into a forward contract to sell 1,782.53 AUD for 1,176.47
USD (1,782.53 x 0.6600)

In two years:
•    1,612.90 AUD grows to 1,612.90e0.05*2 = 1,782.53 AUD
•    1,150.27 USD is needed to repay principal and interest on the
1,000 USD borrowed (1000e0.07*2 = 1,150.27 USD)
•    The forward converts this amount to 1,176.47 USD
•    Riskless profit of 1,176.47 – 1,150.27 = 26.20 USD
Futures on Investment Assets (Commodities)
F0 = S0 e(r+u )T
where u is the storage cost per unit time as a percent of the asset
value (i.e. gold, silver, etc)
Alternatively,
F0 = (S0+U )erT
where U is the present value of the storage costs.

Futures on Consumption Assets
F0 £ (S0+U )erT
– Individuals who keep commodities in inventory do so because
of its consumption value, not because of its value as an
investment
– Ownership of the physical commodity provides benefits that
are not obtained by holders of futures contracts
– As such, we do not necessarily have equality in the equation
Convenience Yield

• The benefit from holding the physical asset
is known as the convenience yield, y
• F0 eyT = (S0 + U)erT , U is the dollar amount
of storage costs
• F0 = S0e(r + u - y)T , u is the per unit constant
proportion of storage costs
The Cost of Carry
• The relationship between futures and spot
prices can be summarized in terms of the cost
of carry
• The cost of carry, c, is the storage cost plus the
interest costs less the income earned
• For an investment asset F0 = S0ecT
• For a consumption asset F0 = S0 e(c–y )T
where, c, is the cost of carry
–   Non dividend paying stock = r
–   Stock index = r – q
–   Currency = r – rf
–   Commodity = r - q + u
Questions (all editions): 5.2, 5.3,
5.4, 5.9, 5.10, 5.14

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 views: 18 posted: 7/22/2013 language: English pages: 20
Lingjuan Ma MS