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:S0:Spot price today F0: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 F 0 = S 0 e rT 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? F 0 = 40e 0.05(0.25) = $40.50

When an Investment Asset Provides a Known Dollar Income F 0 = (S 0 – I )e rT 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? F 0 = ( e -0.03*4/12 )e 0.04*9/12 = $886.60

Arbitrage Opportunities If F 0 > (S 0 – I )e rT, F 0 = $ Action now: -Buy asset$ Borrow$ $39.60 for 4 months at 3% $ for 9 months at 4% -Sell forward for $ In 4 months: -Receive $40 income on asset to pay off the $39.60e 0.03*4/12 = $40.00 first loan with interest In 9 months: -Sell asset for $ Use $860.40e 0.04*9/12 = $ to repay the second loan with interest Profit realized: – = $23.40

Arbitrage Opportunities If F 0 < (S 0 – I )e rT, F 0 = $ Action now: -Short asset to realize$ Invest $39.60 for 4 months at 3% $ for 9 months at 4% -Buy forward for $ In 4 months: -Receive $39.60e 0.03*4/12 = $40.00 interest on investment and pay income of $40 on asset In 9 months: -Buy asset for $ Receive $860.40e 0.04*9/12 = $ from investment Profit realized: – = $16.60

When an Investment Asset Provides a Known Yield F 0 = S 0 e (r–q )T 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 - F 0 is current forward price for a contract that was negotiated some time ago The value of a long forward contract, ƒ, is ƒ = (F 0 – K) e –rT Example (pg 106) Similarly, the value of a short forward contract is ( K – F 0 )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 F 0 = S 0 e (r–q )T where q is the dividend yield on the portfolio represented by the index Example (pg 109)

Index Arbitrage When F 0 >S 0 e (r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F 0 <S 0 e (r-q)T an arbitrageur buys futures and sells (shorts) the stocks underlying the index

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 r f 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 USD per AUD. The two year forward exchange should be: Futures and Forwards on Currencies

Why the Relation Must Be True

Suppose 2-year forward exchange rate is 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, AUD for USD (1, x ) In two years: 620 USD grows to 620e 0.07*2 = USD The 1, AUD is exactly enough to repay principal and interest on the 1,000 AUD borrowed (1000e 0.05*2 = 1, AUD) Need to buy 1, AUD under the forward contract; of the USD, we use USD to do so (696.26/0.6300) Riskless profit of – = USD Arbitrage on Currency Forwards

Suppose 2-year forward exchange rate is USD per AUD Action now: USD is cheaper; Borrow 1,000 USD at 7% per annum for 2 years and convert to 1, AUD at spot exchange rate and invest the AUD at 5% Enter into a forward contract to sell 1, AUD for 1, USD (1, x ) In two years: 1, AUD grows to 1,612.90e 0.05*2 = 1, AUD 1, USD is needed to repay principal and interest on the 1,000 USD borrowed (1000e 0.07*2 = 1, USD) The forward converts this amount to 1, USD Riskless profit of 1, – 1, = USD Arbitrage on Currency Forwards

Futures on Investment Assets (Commodities) F 0 = S 0 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, F 0 = (S 0 +U )e rT where U is the present value of the storage costs. Futures on Consumption Assets F 0  (S 0 +U )e rT –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 F 0 e yT = (S 0 + U)e rT, U is the dollar amount of storage costs F 0 = S 0 e (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 F 0 = S 0 e cT For a consumption asset F 0 = S 0 e (c–y )T where, c, is the cost of carry –Non dividend paying stock = r –Stock index = r – q –Currency = r – r f –Commodity = r - q + u

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