Chapter 3: Determination of Forward and
Future Prices
3.1 Investment Asset vs. Consumption
Asset
Investment asset: asset held for investment
purposes (stocks and bonds)
We can use arbitrage arguments to determine
forward prices from its spot price
Consumption asset: asset held primarily for
consumption (copper, oil)
3.2 Short Selling
Short sale of stocks: sell stocks you don’t own
An investor instructs his broker to short 500
IBM shares: the broker borrows shares from
another client and selling them in the market
If the broker runs out of shares, the investor
has to close out its position
April: short 500 shares IBM at $120 a share
May: dividend of $1 a share is paid to all
shareholders
=> The investor has to pay $500 to the broker
July: the position of the investor is closed out/
buys back the shares at $100 a share
What is the profit of the investor?
3.4 Assumptions and Notation
-
Market Participants
no transaction costs
same tax rate on all net trading profit
borrow and lend money against the same
interest rate
take advantage of arbitrage opportunities
Notation
- T: time until delivery
- S0: price of an asset today
- F0 forward or future price
- r: risk free rate of interest, with continuous
compounding
3.5 Forward rate for an investment asset
(provide no income: zero coupon bonds or
non-dividend paying stocks)
Example: 3.5.1
Risk free rate: 5%
Current stock price: S0 = $40
What is the forward price in three months?
Forward rate: F0 = S0 .e0.05*3/12 = $40.50
F0 = S0 .erT
(3.5)
If the forward price is $ 39
- short one share
- invest the proceeds at the risk free rate
- take a long position in a futures contract
In three month time:
Proceeds grow to 40 .e0.05*3/12 = $40.50
Take delivery of the share
Close out the short position
Net gain: $40.50 - $39 = $1.50
3.6 Known income
(coupon bearing bonds or dividend paying
stocks)
Example:
Coupon bearing bond ($40 in 6 months time
and $40 in one year, just before the end of the
year)
S0 = $900
Six months risk free interest rate: 9 %
One year risk free interest rate: 10 %
I (investment income/ present value):
I = 40 . e -0.09 . 6/12 + 40 . e-0.10 . 1 = 74.433
F0 = (S0 – I) .erT
(3.6)
F0 = (900 – 74.433) e 0.10 .1 = $912.39
3.7 Known yield (this means that the
income is known when expressed as a
percent of the asset price at the time the
income is paid)
F0 = S0 .e (r-q) T
(3.7)
3.8 Valuing Forward Contracts
Delivery date in T years, r is the risk free rate
F0 : current forward price for a contract, that
was negotiated some time ago
K: delivery price in the contract
f: value of a long forward contract today
A general result:
f = (F0 – K) .e-rT
See Example: 3.5.1
T: 3 months
Risk free rate: 5%
S0 = $40
F0 = S0 .erT = $40.50 (= K: delivery price in the
contract)
What is the value of a long forward contract
one month later when the stock price is $42?
F0 = S0 .erT = $42 . e0.05 *2/12 = $42.35
1 month later:
f = (42.35 – 40.50) .e-0.05* 2/12 = $1.83
3.9 Are forward prices and futures prices
equal?
Empirical research:
A number of studies found significant
differences between forward and future prices
It seems likely that these observed differences
are due to:
- transaction costs
- taxes
- treatment of margins
3.10 Stock index futures
A stock index tracks changes in the
hypothetical portfolio of stocks
The weight of a stock equals the proportion of
the portfolio invested in the stock
Future prices of stock indices
A stock index can be regarded as the price of
an investment asset that pays dividends
The investment asset is the portfolio of the
stocks underlying the index and the dividends
are the dividends that would be received by
the holder of the portfolio
F0 = S0 .e (r-q) T
(3.7)
3.11 Forward and future contracts on
currencies
Definition:
S0 and F0 : spot or forward price
in dollars of one unit of the foreign currency
(like the euro, the British Pound or the
Japanese Yen)
The relation between S0 and F0
F0 = S0 .e (r-rf) T
(3.13)
The well known interest rate parity relationship
(courses international economics 1 and
international economics 2)
Example:
S0 = 1.22 USD per euro
r = 0.0375
rf = 0.02
T = ½ year
F0 = S0 .e (r-rf) T = 1.2307
Is it possible for a big financial institution
(lend and borrow money at the risk free rate in
Europe or the United States) to make a profit
if: a) F = 1.24 USD per euro
b) F = 1.22 USD per euro
Show how profits can be made!
3.12 Futures on commodities
Storage costs:
We have seen:
F0 = (S0 – I) .erT
(3.6)
Where: I: present value of investment income
Storage costs can be seen as negative
income:
F0 = (S0 + U) .erT
(3.15)
If storage costs are seen as providing a
negative yield then (3.15) becomes (3.16):
F0 = S0 .e(r+u)T
(3.16)
Convenience yield: (y)
We do not necessarily have equality in
equations (3.15) and (3.16) because users of
a consumption commodity may feel that
ownership of a physical commodity provides
benefits that are not obtained by holders of
future contracts.
The crude oil in inventory (in the presence of
hurricanes like Katrina or Rita) may be
valuable to keep a production process running
and perhaps profiting from temporary
shortages
In this case:
F0 .eyT= (S0 + U) .erT
or
F0.eyT = S0 .e(r+u)T => F0 = S0 .e(r+u-y)T (3.21)
3.13 Cost of carry
Cost of carry (c): storage costs plus the
interest that is paid to finance the asset less
the income earned on the asset
For a non dividend paying stock: c = r
For a stock index: c = r – q
For a currency: c = r – rf
For a commodity with storage costs: c = r + u
For an investment asset
F0 = S0 .ecT
(3.22)
For a consumption asset
F0 = S0 .e(c-y)T
(3.23)
y: the convenience yield
(3.15 Future prices and the expected future
spot price)
To be discussed after knowledge of the
capital asset pricing model (CAPM)
The CAPM tells you all about Risk and
Return.