Determination of Forward and Futures Prices Chapter 5 (all editions)
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Transcript of Determination of Forward and Futures Prices Chapter 5 (all editions)
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Determination of Forward and Futures
Prices
Chapter 5(all editions)
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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)
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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
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Notation
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
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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?
F0 = 40e0.05(0.25) = $40.50
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When an Investment Asset Provides a Known Dollar Income
F0 = (S0 – I )erT
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
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Arbitrage Opportunities If F0 > (S0 – I )erT , F0
= $910.00
Action now:-Buy asset $900.00-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
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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%
-Buy forward for $870.00
In 4 months:-Receive $39.60e0.03*4/12 = $40.00 interest on investment and pay income of $40 on asset
In 9 months:-Buy asset for $870.00-Receive $860.40e0.04*9/12 = $886.60 from investment
Profit realized: 886.60 – 870.00 = $16.60
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When an Investment Asset Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
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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
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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)
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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
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• 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:
Futures and Forwards on Currencies
F S e r r Tf
0 0 ( )
6453.06200.0 )2)(05.007.0(0 eF
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1000 units of foreign currency
at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
1000 units of foreign currency
at time zero
units of foreign currency at time T
Tr fe1000
dollars at time T
Tr feF01000
1000S0 dollars at time zero
dollars at time T
rTeS01000
Why the Relation Must Be True
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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
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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
Arbitrage on Currency Forwards
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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 AssetsF0 (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
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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
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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
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Questions (all editions): 5.2, 5.3, 5.4, 5.9, 5.10, 5.14