Determination of Forward and Futures Prices Chapter 5 (all editions)

1. Determination of Forward and Futures PricesChapter 5(all editions)

2. 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)

3. 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

4. Notation

5. 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

6. 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

7. 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

8. 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

9. When an Investment Asset Provides a Known Yield F0 = S0e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

10. Value of a Forward Contract today • Suppose that -K is delivery price in a forward contract -F0is 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

11. 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 = S0e(r–q )T where q is the dividend yield on the portfolio represented by the index Example (pg 109)

12. Index Arbitrage • When F0>S0e(r-q)Tan arbitrageur buys the stocks underlying the index and sells futures • When F0<S0e(r-q)Tan arbitrageur buys futures and sells (shorts) the stocks underlying the index

13. 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 rfis 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:

14. Why the Relation Must Be True

15. 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

16. 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

17. Futures on Investment Assets (Commodities) F0 =S0e(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

18. 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

19. 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

20. Questions (all editions): 5.2, 5.3, 5.4, 5.9, 5.10, 5.14