Futures and Forwards

1
Futures and Forwards

• September 17, 2001

2
Futures Position

• Enter the contract long or short
• End of contract
• Most do not end in delivery of underlying
• Why not?
• Still we pay attention to delivery details
  why?
• How do the contracts end?

3
Delivery

• If a contract is not closed out before maturity,
  it usually settled by delivering the assets
  underlying the contract. When there are
  alternatives about what is delivered, where it is
  delivered, and when it is delivered, the party
  with the short position chooses.
• A few contracts (for example, those on stock
  indices and Eurodollars) are settled in cash.

4
Futures Contracts

• Exchange traded
• Specifications
• What is deliverable?
• When?
• Where?
• Profit or loss on position is settled on a daily
  basis.

5
Margins

• A margin is cash or marketable securities
  deposited by an investor with his or her broker.
• The balance in the margin account is adjusted to
  reflect daily settlement.
• Margins minimize the possibility of a loss
  through a default on a contract.

6
Example of Futures Trade

• An investor takes a long position in 5 March gold
  futures contracts on December 9th.
• Contract size is 100 oz. (CBOT)
• Futures price is US290.00.
• Initial margin is 5 of contract value.
• Maintenance margin is 75 of initial margin.

7
A Possible Outcome

Daily
Cumulative
Margin
Futures
Gain
Gain
Account
Margin
Price
(Loss)
(Loss)
Balance
Call
Day
(US)
(US)
(US)
(US)
(US)
290.00
7,250
10-Dec
288.60
(700)

(700)

6,550
0
11-Dec
286.00
(1300)
(2,000)
5,250
2,000

7,250

12-Dec
284.80
(600)
(2,600)
6,650
0



15-Dec
286.40
800
(1800)
7,450
0





8
Some Terminology

• Open interest the total number of contracts
  outstanding
• equal to number of long positions or number of
  short positions
• Settlement price the price just before the
  final bell each day
• used for the daily settlement process
• Volume of trading the number of trades in 1 day

9
Regulation of Futures

• Regulation is designed to protect the public
  interest
• Regulators try to prevent questionable trading
  practices by either individuals on the floor of
  the exchange or outside groups

10
Accounting Tax

• If a contract is used for
• Hedging it is logical to recognize profits
  (losses) at the same time as on the item being
  hedged.
• Speculation it is logical to recognize profits
  (losses) on a mark to market basis.
• Roughly speaking, this is what the treatment of
  futures in the U.S. and many other countries
  attempts to achieve.

11
Forward Contracts

• Similar in concept to futures contracts
• Over-the-counter instruments
• Initial value is zero no funds exchanged when
  position established
• No daily marking to market
• Usually ends in delivery of underlying

12
Forward Price

• Forward price is the delivery price applicable if
  the contract were negotiated today.
• The forward price is different for contracts of
  differing maturities.
• What happens to the delivery price over the life
  of a forward contract?

13
Profit from a Long Forward Position
14
Profit from a Short Forward Position
15
Short Selling

• If you short sell, you sell a security you do not
  own.
• The securities are borrowed from another client
  by your broker.
• You owe all cash flows from the security to the
  real owner while you hold the short position.
• Eventually you must buy back the securities and
  return them to the owner.

16
Interest Rates

• Compounding frequency defines the units in which
  interest rates are measured.
• Increasing the compounding frequency increases
  the effective annual rate.
• Once a year
• m times a year

17
Continuous Compounding

• The limit what happens if you compound more and
  more frequently

18
Notation

• Spot price today
• Futures or forward price today
• Time until delivery
• Risk-free rate of interest corresponding to
  maturity T

19
1.Gold An Arbitrage Opportunity?This is slide
23 from last week.

• Suppose that
• The spot price of gold is US390
• The quoted 1-year futures price of gold is US425
• The 1-year US interest rate is 5 per annum
• Is there an arbitrage opportunity?
  

20
2. Gold Another Arbitrage Opportunity?This is
slide 24 from last week.

• Suppose that
• The spot price of gold is US390
• The quoted 1-year futures price of gold is US390
• The 1-year US interest rate is 5 per annum
• Is there an arbitrage opportunity?

21
The Futures Price of Gold slide 25 from last
week

• If the spot price of gold is S the futures
  price is for a contract deliverable in T years
  is F, then
• F S (1r )T
• where r is the 1-year (domestic currency)
  risk-free rate of interest.
• In our examples, S390, T1, and r0.05 so that
• F 390(10.05) 409.50

22
Gold Example

• For gold
• F0 S0(1 r )T
• (assuming no storage costs)
• If r is compounded continuously instead of
  annually
• F0 S0erT

23
Extension of the Gold Example

• Gold is an investment asset.
• For any investment asset that provides no income
  and has no storage costs
• F0 S0erT

24
When an Investment Asset Provides a Known Dollar
Income

• F0 (S0 I )erT
• where I is the present value of the income

25
When an Investment Asset Provides a Known Yield

• F0 S0 e(rq )T
• where q is the average yield during the life
  of the contract (expressed with continuous
  compounding).

26
Stock Index

• Can be viewed as an investment asset paying a
  dividend yield.
• The futures price and spot price relationship is
  therefore
• F0 S0 e(rq )T
• where q is the dividend yield on the portfolio
  represented by the index.

27
Stock Index (continued)

• For the formula to be true it is important that
  the index represent an investment asset.
• In other words, changes in the index must
  correspond to changes in the value of a tradable
  portfolio.
• The Nikkei index viewed as a dollar number does
  not represent an investment asset. It is a
  quanto.

28
Foreign Exchange Quotes

• Futures exchange rates are quoted as the number
  of USD per unit of the foreign currency.
• Forward exchange rates are quoted in the same way
  as spot exchange rates. This means that GBP, EUR,
  AUD, and NZD are USD per unit of foreign
  currency. Other currencies (e.g., CAD and JPY)
  are quoted as units of the foreign currency per
  USD.

29
Futures Forwards on Currencies

• A foreign currency is analogous 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

30
Oil An Arbitrage Opportunity?(slide 26 from
last week)

• Suppose that
• The spot price of oil is US19
• The quoted 1-year futures price of oil is US25
• The 1-year US interest rate is 5 per annum
• The storage costs of oil are 2 per annum
• Is there an arbitrage opportunity?

31
2. Oil Another Arbitrage Opportunity?(slide 27
from last week)

• Suppose that
• The spot price of oil is US19
• The quoted 1-year futures price of oil is US16
• The 1-year US interest rate is 5 per annum
• The storage costs of oil are 2 per annum
• Is there an arbitrage opportunity?

32
Futures on Consumption Assets

• F0 ? S0 e(ru )T
• where u is the storage cost per unit time as a
  percent of the asset value.
• Alternatively,
• F0 ? (S0U )erT
• where U is the present value of the storage
  costs.

33
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 ? S0ecT
• The convenience yield on the consumption asset,
  y, is defined so that F0
  S0 e(cy )T

34
Forward Contracts vs. Futures Contracts
TABLE 2.4 (p. 34)
35
Forward vs. Futures Prices

• Forward and futures prices are usually assumed to
  be the same. When interest rates are uncertain
  they are, in theory, slightly different
• A strong positive correlation between interest
  rates and the asset price implies the futures
  price is slightly higher than the forward price.
• A strong negative correlation implies the reverse.

36
Convergence of Futures to Spot

Futures Price
Spot Price
Futures Price
Spot Price
Time
Time
(a)
(b)
37
Definitions

• Normal market
• Futures price increases with maturity
• Inverted market
• Futures price decreases with maturity
• Normal backwardation
• Contango

38
Futures Prices Expected Future Spot Prices

• Suppose k is the expected return required by
  investors on an asset
• We can invest F0er T now to get ST back at
  maturity of the futures contract
• This shows that
• F0 E (ST )e(rk )T

39
Futures Prices Future Spot Prices (continued)

• If the asset has
• no systematic risk, then k r and F0 is an
  unbiased estimate of ST
• positive systematic risk, then
• k gt r and F0 lt E (ST )
• negative systematic risk, then
• k lt r and F0 gt E (ST )