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:
-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
            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
 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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posted:7/22/2013
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Lingjuan Ma Lingjuan Ma MS
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