Some Notes on Forward Prices
A contract for delivery of an asset at some future date T for a price agreed upon now and
payable at T is a forward contract for the underlying asset. The price you pay at date T
is the forward price, denoted F0,T . The underlying asset may be a commodity such as
corn or crude oil, a financial asset such as a stock or bond, or a rate such as an interest
rate or an exchange rate.
Forwards can be used to create a synthetic version of the underlying asset.
Assume there is a zero coupon bond maturing at date T paying $1.00 face value with
certainty. The price today of this zero is denoted P  0, T  . Enter into a long forward
contract for the asset at the forward price F0,T and buy F0,T zeros. Since there is no cash
flow involved in entering the forward contract, the cost of this strategy is
P  0, T  F0,T .
(1)
At date T , the zeros pay you F0,T which is used to settle the long forward position and
you have the asset. The strategy initiated at date 0 is a synthetic version of the asset, a
prepaid forward contract and (1) gives the prepaid forward price for the asset.
There is another way to view this prepaid forward price using DCF. Let ST
denote the value of the asset at date T and let E0  ST  denote the expectation of this
value at date 0 . The date 0 value of one unit of the asset at date T is a standard
calculation in DCF. Let  denote the appropriate risk-adjusted discount rate for this asset.
Then the date 0 value of receiving one unit of the asset at date T is
eT E0  ST  .
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(2)
Prof. D. C. Nachman
It is important to note that (1) and (2) give the same value, i. e., the date 0 value
of one unit of the asset to be received at date T . Letting r denote the continuously
compounded risk-free rate for the period from 0 to T and equating (1) and (2) gives
e rT F0,T  eT E0  ST  .
(3)
F0,T  e r  T E0  ST  ,
(4)
Rearranging (3) gives
which expresses the forward price as a biased estimate of the expected future spot price
E0  ST  . The bias is due to the risk premium   r .
Expression (3) deserves some emphasis. The date T forward price for the asset
discounted to the present at the risk-free rate is the present value of one unit of the asset
received at time T . If we can observe (or theoretically know) the forward price F0,T , the
prepaid forward price e rT F0,T gives us this present value without having to estimate the
risk-adjusted required rate  or the expected price E0  ST  . This is useful in calculating
the present value of expected future cash flows that depend on future commodity prices.
As a simple example, consider the Kingsley Solomon gold mine example on pp.
278-280 in (BMA). In that example, the production plan for extracting gold calls for
extracting .1 million ounces of gold per year for the next ten years. The value of this plan
now is .1 t 1 PVt , where PVt is the present value of the price of gold in year t . If now
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the ten year forward curve for gold is F0,t , t  1,
,10 , and now the ten year term structure
,10 , then PVt  e rt t Fo,t and the value of the production plan is
is rt , t  1,
.1 t 1 er t F0,t . This is the principle behind the calculations in the Kingsley Solomon
10
t
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example in (BMA), who then use a simplified "spot/futures parity" result to determine the
forward curve F0,t , t  1,
,10 . A more complete picture of this forward curve formation
follows.
This is important even if the forward market for the particular commodity does
not exist if we have a "no arbitrage theory" of the forward price if a market did exist. If a
forward market did exist and the commodity could be leased, as gold can, what would be
the "no arbitrage theory" of the forward price? The commodity lender would be giving up
one unit of the commodity now worth S 0 , the known spot price of the commodity. In
return, at date T the lender gets back elT units of the commodity worth the random
amount ST , where l is the continuously compounded commodity lease rate. The NPV of
this investment is
NPV = eT elT E0  ST   S0
= e
l  T
E0  ST   S0 .
(5)
Suppose the price of the commodity is expected to grow at the continuously compounded
rate of g over the period from 0 to T so that
E0  ST   S0e gT .
Then the NPV of the loan in (5) is
NPV = S0el  g  T  S0 .
(6)
The commodity loan is a fair deal for the lender if the NPV in (6) is zero. This condition
occurs when the lease rate l is set equal to   g , the income yield on the underlying
asset. This competitive lease rate is
l   g .
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(7).
Prof. D. C. Nachman
Note that this lease rate may be positive if the required rate is greater than the expected
growth rate, e. g., if the underlying has a cash yield like a dividend-paying stock. It may
be zero, e. g., if the underlying is a nondividend-paying stock then   g , here shorting
the stock requires only its return as payment. It may be negative for commodities that
have high storage costs. The lease rate is what Hull refers to as net convenience yield, the
convenience yield net of storage costs. See the reference to Hull.
What is the forward price for an underlying with an active lease market? The
lease rate is like a dividend. If we borrow the asset, we have to pay the lender the lease
rate just as if we borrow a dividend-paying stock. By analogy the forward price reflects
this lease rate in that
F0,T  S0e r l T .
(8)
This result is spot/futures parity (see p.732 in (BMA) for a version with discrete
compounding).
Expression (8) gives (BMA) the forward curve for gold in the Kingsley Solomon
example. There (BMA) assume that gold is like a non-dividend paying stock, and
therefore that the lease rate on gold is l  0 . Consequently they get that F0,t  S0ert and
then using (3) they get that PVt  e rt t Fo,t  S0 . The assumption that the lease rate on
gold is zero is just a simplification. Gold can be leased for a positive lease rate. (See the
last two sentences in footnote 6, pp 278-279 in (BMA). More on this in the American
Barrick case.) (8) can be used to infer the lease rate from observed forward prices. If you
know the lease rates then (8) can be used to calculate the forward curve.
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Reference:
th
Brealey, R. A., Myers, S. C., and Allen, F. 2006. Principles of Corporate Finance, 8 ed.,
McGraw- Hill/Irwin, Inc.
McDonald, R. L., 2003. Derivatives Markets. Boston: Addison Wesley. The material here
was taken from chapter 6.
Hull, J. C., 2003. Options, Futures, and Other Derivatives, 5th ed. Prentice Hall. (3.21) p.
60.
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Prof. D. C. Nachman