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?

F0 = 40e0.05(0.25) = $40.50

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

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

F0 = S0 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

-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

• 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

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

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

Arbitrage on Currency Forwards

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

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