Introduction

• In the previous chapter, it was shown that the price of a futures contract and the price of a forward contract with the same underlying asset and maturity are approximately equal. This is true whether the asset is a commodity or a financial asset. This chapter will therefore treat futures as forwards (i.e., the daily settlement will be ignored).
• Most commodities are consumption assets. This means that they are rarely held for purely investment reasons (metals such as gold and silver are exceptions). Commodity owners usually intend to use the commodity in some way, after which it ceases to be available for sale.
• For most commodities, no-arbitrage arguments can only be used to obtain an upper bound for the futures price. An additional unobserved parameter, known as the convenience yield, is required to determine the commodity futures price in the market.

Commodities Versus Financial Assets

• There are several important differences between commodities and financial assets. Some differences in particular are as follows.

1. Storage Costs – The storage costs associated with financial assets (e.g., stocks and bonds) are negligible. The storage costs for commodities, however, can be quite substantial. These costs include insurance, which can vary over time. Some commodities deteriorate with time and require special (i.e., costly) care in storage. Assets like corn and natural gas are frequently stored for use at a particular time of the year. Other assets, like oil and copper, are consumed throughout the year.
2. Transport – Commodities can be costly to transport and thus their prices can depend on their location. By contrast, financial assets are usually transported electronically at virtually no cost.
3. Lease Rate – A commodity held for investment purposes (e.g., gold or silver) can be borrowed for shorting by paying a lease rate. This lease rate can exceed the fees charged when financial assets are borrowed for shorting.
4. Expected Return and Mean Reversion of Prices – A financial asset provides investors with an expected financial return that reflects its risk. Most commodities do not have this property. Indeed, it can be argued that the prices of most commodities are mean reverting. This means that although the price of a commodity can be quite volatile, it tends to get pulled back toward some central value.
• When the price of a commodity is relatively high, its production will become attractive and its supply will therefore tend to increase. Also, users of the commodity may search for less expensive alternatives. These actions will tend to reduce the price.
• If the price is at a relatively low level, on the other hand, the production of the commodity will become less attractive and supply will reduce, while its use will tend to increase. As a result, its price will tend to rise.

• As a result of these distinctions, the futures prices of commodities can behave quite differently from those of financial assets.

Types of Commodities – Agriculture Commodities

• Agricultural commodities with futures contracts include products that are grown (e.g., corn, wheat, soybeans, cocoa, and sugar) as well as livestock (e.g., cattle and hogs). It is expensive to store agricultural commodities; and some products can only be stored for a limited period of time. There can be interdependence among various agricultural commodities. For example, cost of feeding livestock can depend on prices of commodities like corn.
• The prices of agricultural commodities can be seasonal. For example, at harvesting time (October to November), prices of corn and soybeans tend to be lower. During other months, prices may be higher due to storage costs. This seasonality is sometimes reflected in futures prices, causing them to display a mixture of normal and inverted pricing patterns. There are also other factors that may affect the market’s view on the future price
• If there has been (or if there is expected to be) a good (or bad) harvest, market participants will expect prices to be relatively low (or relatively high).
• Political considerations can also affect futures prices.
• Weather is also an important consideration for many agricultural commodities. Bad weather in Florida can increase the futures price of frozen orange juice. Frosts in Brazil are liable to drastically reduce Brazilian coffee production and increase coffee futures prices.

Types of Commodities – Metals

• Commodity metals include gold, silver, platinum, palladium, copper, tin, lead, zinc, nickel, and aluminum. Their properties are quite different from agricultural products. For example, metal prices are not affected by the weather and are not seasonal (because they are extracted from the ground). Additionally, their storage costs are typically lower than those of agricultural products.
• Some metals are held purely for investment purposes. This means that their futures price can be more easily obtained from observable variables. For someone looking to hold a metal for investment, owning a futures contract can be an acceptable alternative to owning the physical asset itself.
• Inventory levels are important in determining prices. Most metals are extracted in one country and consumed in another. As a result, exchange rates may affect prices. Other crucial factors include the scope of a metal’s industrial application and the rate at which new sources are found. Changes in extraction methods, actions by governments (and/or cartels), and environmental regulations can also impact metal prices. Sometimes metal prices can even be affected by recycling processes. For example, a metal used in a production process one year can be recycled and re-enter the market many years later.

Energy

• There are futures contracts on crude oil and crude oil extracts (e.g., petroleum and heating oil). The crude oil market is the largest commodity market in the world and global demand is estimated to be nearly 100 million barrels per day. There are many grades of crude oil reflecting variations in gravity (density) and sulfur content. Two important benchmarks are Brent crude oil (which is sourced from the North Sea) and West Texas Intermediate (WTI) crude oil.
• Electricity is an unusual commodity because it is almost impossible to store. The non-storability of electricity can lead to huge price swings. For example, heat waves in the summer can drive up the demand for air conditioning and have been known to increase the cost of electricity by as much as 1,000%. Once the heat wave is over, however, the price quickly returns to normal levels. Futures contracts on electricity are less actively traded than futures contracts on natural gas and crude oil. They also trade in the over the counter market. A typical contract allows one side to receive a specified number of megawatt hours for a specified price at a specified location during a particular month. In a 5 × 8 contract, power is received during the off-peak period (11 p.m. to 7 a.m.) from Monday to Friday. In a 5 × 16 contract, power is received during the on-peak period (7 a.m. to 11 p.m.) from Monday to Friday for the specified month. In a 7 × 24 contract, power is received 24 hours a day, seven days a week for the specified month.

• Natural gas is used for heating and generating electricity. It can be stored (either above ground or underground) for indefinite periods, but its storage costs are quite high. Natural gas is also expensive to transport, and hence the price of natural gas can vary regionally. CME Group offers a futures contract on 10,000 million British thermal units (BTUs) of natural gas. The contract requires delivery to be made at a roughly uniform rate to a hub in Louisiana. Intercontinental Exchange (ICE) also offers a futures contract on natural gas. The demand for natural gas is high in the winter (for heating purposes) and to a lesser extent the summer (to produce electricity for air conditioning). This creates seasonality in the futures prices, and generally winter month contracts are priced a little higher.

Weather

• Derivative contracts on weather are available in both the exchange-traded and over-the counter markets. The most popular contracts are those with payoffs contingent on temperature (which are used by energy companies as hedges). Two important weather derivative variables are heating degree days (HDDs) and cooling degree days (CDDs). The HDD and CDD for a day are (respectively) defined as:

HDD = max⁡(0, 65 – A)

and

CDD = max⁡(0, A – 65)

where

A is the average of the highest and lowest temperature during a given day at a specified weather station (as measured in degrees Fahrenheit).

For example, if the minimum temperature during a day (midnight to midnight) is 40 degrees Fahrenheit and the maximum is 60 degrees Fahrenheit,

The daily CDD is therefore zero, and the daily HDD is 15. Contracts are usually defined in terms of the cumulative HDD and CDD for all the days in a given month.

Commodities Held for Investment

• Some precious metals are held for investment purposes. Gold and silver (and to a lesser extent platinum and palladium) are in this category. While these metals have industrial applications, some purchasers hold these commodities purely for investment purposes. For them, owning a futures or forward contract can be a practical substitute for owning the commodity itself.
• The storage costs of the metals held for investment are generally low compared to metal prices and can therefore be ignored. Additionally, there is a generally small lease rate associated with metals held for investment. For example, gold is like a financial asset in that it can be lent from one entity to another to earn interest.
• In analyzing the futures prices for investment commodities, the lease rate is ignored first. This means that the relationship between the forward price and the spot price is

where

T is the time to maturity of the forward (measured in years) and

R is the annually compounded risk-free interest rate for maturity T.

• The no-arbitrage arguments are as follows.
  • Arbitrage A: If the forward price is greater than
    , a trader can buy the investment commodity at price S (by financing the purchase at rate R) and at the same time enter into a forward contract to sell it at time T.
  • Arbitrage B: If the forward price is less than
    , a trader who owns the investment commodity will find it profitable to sell it at price S and at the same time enter into a forward contract to buy it back at time T.

Lease Rates

• The lease rate for an investment commodity is the interest rate charged to borrow the underlying asset.

For example, a gold producer might enter into a contract with an investment bank to sell its future production forward at a predetermined price. The bank will then hedge the risk it is taking on (with respect to the price of gold) by borrowing gold for a time period equal to the life of the forward contract and selling it in the spot market. This synthetically creates a short forward contract that offsets the contract it has with the gold producer. The main lenders of gold are central banks.

Let
be the lease rate. The relationship between the forward price and the spot price is

This equation can be used to calculate an implied lease rate:

• Example:

Assume that the spot price of gold is USD 1,240, the six-month futures price is USD 1,250, and the six-month risk-free rate is 4% (with annual compounding).

• The lease rate of gold varies with the supply of gold that can be borrowed along with the demand to borrow gold. Recall that when gold producers hedge future production, the banks on the other side of the transaction borrow gold. As gold producers hedge more (less), the demand for borrowed gold will increase (decrease) and the lease rate will rise (fall). Similarly, as asset owners become more (less) willing to lend gold, the lease rate will tend to fall (rise). Occasionally the lease rate is negative and may therefore allow arbitrageurs to buy the metal and sell it forward for a profit.

Convenience Yields

• Now lets consider consumption commodities (e.g., crude oil, copper, and corn). Assume that there are no storage costs. If
  , traders can undertake arbitrage type A. Traders can buy the asset by financing the purchase at rate R and entering a forward contract to sell it at price F at time T. Therefore, the upper bound to the forward price is , i.e.

• When the storage costs with present value U are considered, the arbitrageur has extra costs to finance, and there is an extra repayment of
  
  required at time T. This means that

• The convenience Y balances this equation to get the equality from the inequality

so that

• The convenience yield measures the benefits to the asset holder of having it in their inventory as a protection against future shortages or delivery delays.
  • Suppose there is expected to be a plentiful supply of the asset during the life of a futures contract. This means it can be ordered at any time for almost immediate delivery. In this case, the convenience yield is likely to be close to zero and

might be a reasonable approximation.

• If inventory levels of the asset are low and there are concerns about shortages, however, the convenience yield will likely be quite high. In that case, it is most likely the case that

Convenience Yields – Example

• Suppose that the spot price of oil is USD 69 per barrel, and the six-month futures price is USD 65 per barrel. The cost of storing oil for six months has a present value of USD 1 per barrel, and the risk-free rate is 2% per year.

One way of interpreting this number is to view it as the cost of borrowing oil (if that were possible). Because physical oil provides benefits to the holder at the rate of 18.3% per year, this is arguably the rate that would be charged to borrow it. it appears that the market was expecting the price of oil to decline over the six-month period. Under such circumstances, holding oil as an investment makes no sense. (Indeed, if oil were an investment asset, its price would reflect market sentiment.) The only reason someone would hold oil is to use it. An expectation that the price of an asset will decline sharply is therefore consistent with the physical asset having a high convenience yield.

Cost of Carry

• The cost of carry for an asset reflects the impact of:
  • Storage costs,
  • Financing costs, and
  • Income earned on the asset.
• In the case of a financial asset, there are no storage costs. If the financing cost is R and the yield is Q (both expressed with annual compounding), the cost of carry per year is

which is often simply approximated as R – Q.

• Example:
  • Consider a foreign currency. If the domestic interest rate is 4% and the foreign interest rate is 3%, the cost of carry is approximately 1%. If the foreign interest rate were 6%, the cost of carry would be around −2% (this is referred to as a negative carry).

• In the case of a commodity, there is usually no income. The cost of carry therefore consists of the interest rate and the storage cost. In the previous example

Continuous Versus Discrete Compounding

• When R and Q are expressed with continuous compounding, the forward price formula in becomes

• When the storage cost (as a fraction of the asset price) is expressed with continuous compounding, the cost of carry for a consumption commodity is the interest rate plus the storage cost. The relationship between the futures price and the spot price is
  

where

C is the cost of carry, and

Y is the convenience yield

(both expressed with continuous compounding).

• When the annually compounded rate is low, it more closely resembles the equivalent continuously compounded rate and thus it is often reasonable to ignore compounding frequency issues. As the rate increases in magnitude, there is a greater approximation error in treating these two rates as equivalent.

Expected Future Spot Prices

• The expected future spot price of an asset is the market’s average opinion about what the spot price will be in the future. Natural questions are as follows.
  • Does the futures price of an asset equal the expected future spot price?
  • Is the futures price a good forecast of the future spot price?
• The futures price converges to the spot price at maturity of the contract. If an investor thinks the spot price at maturity will be greater than the current futures price, the investor can take a long futures position. Similarly, if an investor thinks the spot price at the maturity will be less than the current futures price, the investor can take a short futures position. In either case, an investor that is correct will be able to close out the futures contract near the maturity for a profit.
• These trading strategies do not involve storing the commodity or investing in a physical asset. They only involve trading futures contracts. In the example where the six-month futures price of oil is USD 65, investors who think this is too low will take long futures positions, and investors who think it is too high will take short futures positions.

• John Maynard Keynes argued that speculators require compensation for the risks they are bearing, whereas hedgers derive benefits from futures and thus do not require any such compensation. Indeed, the hedgers might be prepared to lose money (on average) because their overall market risks are reduced by hedging. If hedgers tend to hold short positions, and speculators tend to hold long positions, Keynes’ argument suggests the longs will tend to make money on average and the shorts will tend to lose money on average. The futures price should therefore be less than the expected future spot price. If the reverse is true (i.e., hedgers tend to hold long positions and the speculators hold short positions), the shorts will make money on average, and the futures price will be greater than the expected future spot price.
• More recent work on this subject has involved the capital asset pricing model (CAPM). Using CAPM, it is argued that an investor should earn a return greater than the risk-free rate when the systematic risk of his or her portfolio is positive. Suppose that P is the present value of the futures price when discounted from time T to time zero at the risk-free rate:

The trader can create a long position in the asset at time T by:

• • investing P at the risk-free rate, and
  • Entering into a long futures contract to buy the asset for F at time T.

Let

be the spot price at time T. The cash flows from the trader’s strategy are

Time 0 : – P , and

Time T: +

Let E denote the expected value operator. Thus, the expected cash flow at time T is
. Let the expected return from the trades which were just considered be X Hence

Substituting P gives

 

This shows that the relationship between

and F depends on X and R.

• • If X > R, then
    
    > F
  • If X < R, then
    
    < F

• The variable X is the return from the synthetically produced position. The systematic risk of this investment depends on the correlation between the underlying asset return and the market return .
  • If this correlation is positive, the systematic risk of the investment will be positive (which implies X > R and therefore
    
    > F).
  • If the correlation is negative, the systematic risk of the investment is negative (which implies X < R and thus
    
    > F).
  • When the correlation is zero, the synthetic trade has no systematic risk, and the futures price should be equal to the expected future spot price.

These results apply to the FX forwards and futures and financial futures as well considered in this chapter. They are summarized in the table in the next slide

Correlation Type	Systematic Risk	X Versus R
Underlying asset return and the market return are uncorrelated.	Synthetic trade has no systematic risk.	X = R	Futures price equals expected future spot price.
Underlying asset return is positively correlated with the market return.	Synthetic trade has positive systematic risk.	X > R	Futures price is less than expected future spot price.
Underlying asset return is negatively correlated with the market return.	Synthetic trade has negative systematic risk.	X < R	Futures price is greater than expected future spot price.

Relationship Between Futures Price and Expected Future Spot Price as suggested by CAPM

• An example of an asset with positive systematic risk is the S&P 500 itself. It can be, therefore expected that
  . It is known the forward/future price of a stock index is given by:

where

Q is the dividend yield on the index.

• Suppose that the index equals the forward price at time T. If dividends are reinvested in the index, the position in the index will grow at rate Q, and value of the investment at time T is

This shows that the return realized by the investor is the risk-free rate (R). However, because the index (by definition) is positively correlated to itself, the investor should expect to earn more than R. This means that the expected value of the index at time T should be greater than F.

Modern Theory

• Many commodities have positive systematic risk because they tend to cost more when the economy is doing well. An exception may be gold, which is often referred to as having negative systematic risk. When the economy is doing poorly, investors increase their holding of gold, and its price increases. When the economy starts to recover, gold is exchanged for stocks, and its price declines. Assuming this is true, theory suggests that the futures price of gold overstates the expected future spot price:

• For those assets with little or no systematic risk, CAPM argues that the futures price should be equal to the expected future spot price. The futures price probably provides the best available forecast of the spot price in this case. However, different industry experts could reasonably come up with different forecasts, and it is difficult to empirically determine the best forecast.

Backwardation and Contango

• When the futures price is below the expected future spot price, the situation is known as normal backwardation. When the futures price is above the expected future spot price, the situation is known as normal contango.
• The term backwardation is used to refer to the situation where the futures price is below the current spot price. Contango is then used to refer to the situation where the futures price is above the current spot price.