Chapter 6 Commodity Forwards and Futures. Copyright © 2006 Pearson Addison-Wesley. All rights...

Chapter 6

Commodity Forwardsand Futures

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-2

Introduction to Commodity Forwards

• Commodity forward prices can be described by the same formula as that for financial forward prices

F0,T S0e(r )T


Introduction to Commodity Forwards (cont’d)

• For financial assets, is the dividend yield

• For commodities, is the commodity lease rate The lease rate is the return that makes an investor

willing to buy and then lend a commodity The lease rate for a commodity can typically be

estimated only by observing the forward prices


Introduction to Commodity Forwards (cont’d)

• The set of prices for different expiration dates for a given commodity is called the forward curve (or the forward strip) for that date

• If on a given date the forward curve is upward-sloping, then the market is in contango. If the forward curve is downward sloping, the market is in backwardation

Note that forward curves can have portions in backwardation and portions in contango


Equilibrium Pricing of Commodity Forwards

A long commodityforward contract at theprice F0,T.

+

0 ST F0,T

A zero-coupon bondthat pays F0,T at time T.

e rT F0,T F0,T

Total e rT F0,T ST = the value of a unit of

the commodity at time T.

• A synthetic commodity can be created by combining a forward contract with a zero-coupon bond

Investment strategy: Cost at time 0: Payoff at time T:


F0,t E0 (ST )e(r )T

Equilibrium Pricing of Commodity Forwards (cont’d)

• As with financial forwards, the commodity forward price is a biased estimate of the expected spot price, E0(ST ), with the bias due to the risk premium on the commodity, -r


Equilibrium Pricing of Commodity Forwards (cont’d)

• Different commodities have their distinct forward curves, reflecting different properties of

Storability Storage costs Production Demand


Nonstorability: Electricity

• Electricity has the following characteristics It cannot be easily stored. Therefore, it is not possible to

engage in arbitrage At any point in time, the maximum supply of electricity is fixed Demand for electricity varies substantially by season, by day

of week, and by time of day

• Given these characteristics, electricity forwards have large price swings over the day. Price swings reflect changes in the expected spot price, which in turn reflects changes in demand over the day


Pricing Commodity Forward by Arbitrage: An Example

• Suppose that pencils cost $0.20 today and, for certainty, will cost $0.20 in 1 year

• Suppose that the continuously compounded interest rate is 10%.

• Since a lender of the pencil has invested $0.20 in a pencil, he will require a borrower to pay interest. Therefore, the pencil has a lease rate of 10%

• It can be demonstrated that the pencil forward price must be $0.20


Pricing Commodity Forward by Arbitrage: An Example (cont’d)

• Any forward price less than $0.20 results in arbitrage profits

• Any forward price greater than $0.20 results in arbitrage profits


Pricing Commodity Forward by Arbitrage: An Example (cont’d)

• Using no-arbitrage arguments, we have demonstrated that the pencil forward price is $0.20

• Verify that

where , the discount rate for a risk-free pencil, is equal to r, the risk-free rate

F0,T E0 (ST )e(r )T 0.20 e(.10 .10) 0.20


The Commodity Lease Rate

• The lease rate is the difference between the commodity discount rate, , and the expected growth rate of the commodity price, g

• For a commodity owner who lends the commodity, the lease rate is like a dividend

With the stock, the dividend yield, , is an observable characteristic of the stock

With a commodity, the lease rate, l, is income earned only if the commodity is loaned. It is not directly observable, except if there is a lease market

l g


Forward Prices and the Lease Rate

• The lease rate has to be consistent with the forward price

• Therefore, when we observe the forward price, we can infer what the lease rate would have to be if a lease market existed

• The annualized lease rate

• The effective annual lease rate

l r 1

TIn (F0,T / S)

l (1 r)

(F0,T / S)1/T 1


The Commodity Lease Rate

• Contango occurs when the lease rate is less than the risk-free rate

• Backwardation occurs when the lease rate exceeds the risk-free rate


Carry Markets

• A commodity that is stored is said to be in a carry market

• Reasons for storage

There is seasonal variation in either supply or demand (e.g., some agricultural products)

There is a constant rate of production, but there are seasonal fluctuations in demand (e.g., natural gas)


Storage Costs and Forward Prices

• One will only store a commodity if the PV of selling it at time T is at least as great as that of selling it today

• Whether a commodity is stored is peculiar to each commodity

• If storage is to occur, the forward price is at least

where (0,T) is the future value of storage costs for one unit of the commodity from time 0 to T

F0,T S0erT (0,T )


Storage Costs and Forward Prices (cont’d)

• When there are storage costs, the forward price is higher. Why?

The selling price must compensate a commodity holder for both the financial cost of storage (interest) and the physical cost of storage

• With storage costs, the forward curve can rise faster than the interest rate


Storage Costs and the Lease Rate

• If there is a carry market for a commodity, what is the lease rate in this case?

• The lease rate should equal the negative of the storage cost. Why?

If one lends a commodity, he is saved from having to pay storage costs. That is, the commodity borrower is providing “virtual storage” for the commodity lender, who receives back the commodity at a point in time. The lender will pay the borrower!


The Convenience Yield

• Some holders of a commodity receive benefits from physical ownership (e.g., a commercial user)

• This benefit is called the commodity’s convenience yield

• The convenience yield creates different returns to ownership for different investors, and may or may not be reflected in the forward price


The Convenience Yield (cont’d)

• From the perspective of an arbitrageur, the price rage within which there is no arbitrage is

where c is the continuously compounded convenience yield

• The convenience yield produces a no-arbitrage range rather than a no-arbitrage price. Why?

There may be no way for an average investor to earn the convenience yield when engaging in arbitrage

S0e(r c)T F0,T S0e

(r )T

Chapter 6 Commodity Forwards and Futures. Copyright © 2006 Pearson Addison-Wesley. All rights...

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