Chapter 6 Commodity Forwards and 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-3 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-4 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-5 Equilibrium Pricing of Commodity Forwards 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:

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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-7 Equilibrium Pricing of Commodity Forwards (cont’d) Different commodities have their distinct forward curves, reflecting different properties of  Storability  Storage costs  Production  Demand

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-8 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-9 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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)

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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!

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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

Copyright © 2006 Pearson Addison-Wesley. All rights reserved 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