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Introduction commodity spot and futures markets


  • Bill of lading is a document that mentions the commodity owner and acknowledges that the goods have been received as cargo and are ready for delivery
  • The major risks involved with commodity transactions are:
  • Price risk:  Risk of downward movement in price. Futures/Forward contracts reduce this      risk
  • Transportation risk: Consists of two risks:
  • Ordinary: Deterioration, spoilage, accident etc.
  • Extraordinary: wars, riots, strike etc.
  • Delivery risk: Parties may withdraw from delivery. This risk has been greatly decreased by robust practises by clearing houses
  • Credit risk: Counter party risk which is mainly an issue in spot market
  • Commodity markets also, like financial markets consists Hedgers, Speculators and Arbitrageurs
  • Hedgers are generally farmers/ranchers who want to lock in a price

Basis risk in commodity futures

  • Basis is the difference between spot price and the price of commodity’s future contract at any given time
  • Changes in basis is due to changes in cost of carry of the asset. Basis risk is generally represented by the volatility / variance of the basis over time
  • σ2S(t)-F(t) = σ2S(t) + σ2f(t) - 2σS(t)σf(t) ρs,f 
  • Hedge effectiveness =

Commodity Forwards

  • Commodity forward prices can be described using the same formula as used for financial forward prices
  • For financial assets, δ is the dividend yield
  • For commodities, δ is the commodity lease rate
  • The lease rate is the return that makes an investor willing to buy and lend a commodity
  • Some commodities (metals) have an active leasing market
  • Lease rates can typically only be estimated by observing forward prices

Futures term structure

  • 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


  • Since r is always positive, assets with δ =0 display upward sloping (contango) futures term structure
  • With δ >0, term structures could be upward or downward sloping

A commodity loan

  • If you loan a commodity, you are giving up S0 today, and will get back St
  • If loan is fairly priced, its NPV = 0
  • NPV = E0(St)e-αT – S0
  • Where α is required return on the commodity
  • Now, suppose commodity price grows at rate g, E0(ST)= S0egT
  • Then, NPV = S0e(g-α)T – S
    • If g<α, NPV<0
    • this is common for commodities (supply with near-perfect elasticity)
    • Therefore, to make loan feasible, you would require lender to pay you the α-g difference
    • This would get NPV back to zero
    • If 1 unit is loaned, you receive a lease payment of e(α-g)T units, and NPV = 0

The Commodity Lease Rate

  • The lease rate (δ) is the difference between the commodity discount rate, α, 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, δ, is an observable characteristic of the stock
  • With a commodity, the lease rate, δ l, is income earned only if the commodity is loaned. It is not directly observable unless there is an active lease market

  • With the addition of the lease payment, NPV of loaning the commodity is 0
  • The lease payment is like the dividend payment that has to be paid by the person who borrowed
  • a stock
  • Therefore:


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

Storage and Carry Markets

  • A commodity that may be 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 λ(0,T) is the future value of storage costs for one unit of the commodity from time 0 to T


Storage Costs and Forward Prices (cont’d)

  • When there are storage costs, the forward price is higher. Why?
  • The forward price must compensate a commodity holder for both the financial cost of carry (interest) and the physical cost of carry (storage)
  • With storage costs, the forward term structure can be steeper than the interest term structure
  • 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
  • Convenience and leasing
    • If someone lends the commodity they save storage costs, but lose the ‘convenience’
    • Stated as (λ –c)
    • Therefore, commodity borrower pays a lease rate that covers the lost convenience less the storage costs:
    • δ = c – λ 

Pricing with convenience

  • So, if:


  • And if, δ = c – λ 
  • Then,  F0,T  = S0e(r+ λ -c)T

No-Arbitrage with Convenience

  • From the perspective of an arbitrageur, the price range 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


Suppose that the price of corn is $2.20/bushel, the effective annual interest rate is 4.6%, and effective annual priced storage costs are 10% of the current price/bushel.  What is the 6-month forward price?


F = 2.2e (.046 + 0.1) 0.5 = $2.37
Now suppose the holder of the asset realizes a convenience yield of 2%. What is the price?
F = 2.2e (.046 + 0.1 - 0.02) 0.5 = $2.34
The futures price dropped because the cost of carrying corn dropped


Hedging oil costs?

  • Suppose we are scheduled to purchase 15,000 bbls of oil in July 2008. The current futures price is $105/bbl, and each contract covers 1,000 bbls.  If we hedge, what is our cost of oil if the spot price of oil in July 2008 is $70/bbl or $120/bbl?
    • Our natural exposure is short, therefore hedge long
    • Direct hedge, β = 1.  N=15/1*1 = 15 contracts
    • Payoff = -15,000ST + 15*1000*(ST – 105)
    • Payoff = -$1,575,000, at any future oil ST

Hedging production costs

  • Suppose oil is a major component of our total production costs which equal $40 million, but it is not the only component.  In general, our production costs rise/fall with sensitivity of 0.72 (beta=0.72) to oil.  Each crude oil contract is on 1,000bbls. Suppose S0=108 and F=105.
    • Now, how many contracts do we use to hedge?
    • Cross hedge, β = 0.72.  N = 40m/105,000*0.72 = 274.28 contracts
    • Suppose oil goes up by 10% from 108 to 118
      • Increase in production costs = 0.072*40 = $2.88 million
      • Payoff from forwards = 274.28*1,000 (118-105) = $2.88 million
  • Note, we now have basis risk – the basis for our hedge does not match the hedging instrument perfectly – what if our relation is not 0.72?

Strip and Stack Hedges

  • In the last example, we bought 450K bbls forward
  • This might be one component of a “strip hedge” if we are selling forward in other periods as well
  • In a “stack hedge”, we enter near-term contracts sufficient to cover the present value of future obligations
  • We then “roll the hedge” into new contracts as the near-term contracts expire

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