FMP 11: Commodity Forwards and Futures
On paper a forward contract on a barrel of oil and a forward contract on a share of stock look like the same animal. The pricing logic behind them is not the same, and the gap opens up because the two underlying assets are owned for entirely different reasons. Daily settlement is set aside throughout, so futures contracts and forward contracts are treated as interchangeable, which is a reasonable simplification for a commodity just as it is for a financial asset.
Most commodities are consumption assets. They are rarely bought as a pure investment, metals such as gold and silver being the standing exception, and the buyer normally means to burn, feed, refine or mill the thing, after which it leaves the market. The no-arbitrage machinery built for financial assets assumes an owner indifferent between the asset and a claim on it, and a consumption asset owner is not indifferent at all.
Four practical differences
Storage costs on stocks and bonds round to nothing. Storage costs on physical goods can be large, they include insurance whose price moves around, and several commodities degrade unless they are looked after expensively. Corn and natural gas are laid down deliberately for use at one season of the year, while oil and copper are drawn on all year round.
Transport is the second difference, since moving a cargo of ore or grain costs enough that the price is partly a fact about where the goods sit, whereas a bond travels between accounts electronically. Borrowing is the third: a commodity held for investment can be borrowed by a short seller, who pays a lease rate that can run above the fee on a borrowed financial asset.
Expected return is the fourth. A financial asset offers investors a return matched to its risk, while most commodities offer nothing of the kind, and a good case can be made that their prices are mean reverting instead: volatile in the short run, yet tugged back toward a central value. A high price makes production attractive and sends buyers hunting for substitutes, which presses the price down again, and a low price does the reverse.
Add these together and commodity futures prices behave quite unlike financial futures prices, in that the no-arbitrage argument yields only a ceiling rather than a single value, and an unobservable extra parameter, the convenience yield, is needed to explain the traded price.
Agricultural commodities
Commodities with liquid futures markets fall into three families. The agricultural group covers things that are grown, such as corn, wheat, soybeans, cocoa and sugar, alongside livestock such as cattle and hogs. Keeping these goods is expensive, and even in ideal conditions many survive only a limited time in a warehouse. The group is knitted together as well, since the cost of feeding livestock depends on the price of grown crops such as corn.
Prices in this family follow the calendar. Corn and soybeans are cheapest around the harvest, October to November, and firmer through the rest of the year as farmers and distributors carry the crop and pay for the privilege. Futures curves inherit that pattern, mixing normal and inverted shapes along one curve rather than following a single clean slope.
Expectations about the next harvest move the curve as well, downward when the crop looks abundant and upward when it looks poor, and politics enters directly: trade tensions between the United States and China in 2018-2019 made exporting corn and soybeans harder for American farmers. Weather is the other permanent factor. A cold snap in Florida lifts the futures price of frozen orange juice, and frosts in Brazil can cut the coffee crop enough to send coffee futures sharply higher.
Metals
The metals group runs from gold, silver, platinum and palladium through copper, tin, lead, zinc, nickel and aluminum. Metals are dug out of the ground rather than grown, so weather barely touches them, their prices are not seasonal, and storage is cheap relative to the value stored.
Some metals are bought purely as an investment, which matters for pricing, because their futures price can be pinned to observable quantities and an investor can hold the contract instead of a vault position without giving up anything essential.
Inventory still governs the short run, exactly as it does for crops. Metals are usually mined in one country and consumed in another, which brings exchange rates into the price, and beyond that prices respond to the breadth of industrial uses, the pace at which new deposits are proved up, extraction technology, actions by governments and cartels, and environmental rules. Recycling adds a long tail of supply, since a metal put into a product one year can be melted down and sold again many years afterwards.
Energy is the third family, carrying futures on crude oil, on refined products such as petroleum and heating oil, on natural gas and on electricity.
Crude oil supports the largest commodity market anywhere, with world demand put at roughly 100 million barrels per day. Crude is not one substance: grades differ in gravity, meaning density, and in sulfur content, and two benchmarks dominate the quotes, Brent crude oil from the North Sea and West Texas Intermediate, shortened to WTI.
Natural gas heats buildings and generates electricity. It can be held above ground or underground indefinitely, although storage is costly, and transport is expensive enough that its price differs by region. The CME Group contract covers 10,000 million British thermal units, delivered at a roughly even rate to a hub in Louisiana, and Intercontinental Exchange, known as ICE, lists a contract of its own. Demand peaks in winter for heating and again, less sharply, in summer when air conditioning lifts power generation, and that two-humped pattern shows up in the futures curve.
| Maturity month | Settlement price | Change from month above |
|---|---|---|
| February 2020 | 1.926 | |
| April 2020 | 1.940 | +0.014 |
| July 2020 | 2.141 | +0.201 |
| October 2020 | 2.199 | +0.058 |
| January 2021 | 2.609 | +0.410 |
| April 2021 | 2.255 | -0.354 |
| July 2021 | 2.306 | +0.051 |
| October 2021 | 2.320 | +0.014 |
| January 2022 | 2.672 | +0.352 |
| April 2022 | 2.259 | -0.413 |
Source: CME Group natural gas futures settlement prices, quoted on January 24, 2020. The third column is calculated here.
Electricity is the awkward member of the family, because storing it is very nearly impossible. Indirect routes exist, among them running a hydroelectric plant in reverse to lift water into its reservoir for release later, but nothing that amounts to a warehouse, so what a region can supply at any moment is capped by the generating capacity inside it. The United States is divided into 140 such regions, each setting its own wholesale price, and sales across a boundary depend on whether transmission lines exist and what they can carry.
Without storage nothing smooths a shock, so prices swing violently. Summer heat waves have on occasion lifted electricity costs by as much as 1,000%, after which the price drops straight back to normal, which is mean reversion in its most vivid form.
Electricity futures do trade, far less actively than natural gas or crude oil futures, and similar contracts trade over the counter. A typical contract delivers an agreed number of megawatt hours at an agreed price, at a named location, across a stated month, with the delivery window written into the contract name. A 5 X 8 contract supplies the off-peak block, 11 p.m. to 7 a.m., Monday through Friday. A 5 X 16 contract supplies the on-peak block, 7 a.m. to 11 p.m. A 7 X 24 contract runs around the clock. Prices across these blocks are strongly seasonal, with summer demand the driver.
Weather derivatives
Contracts written on the weather itself trade on exchanges and over the counter, and the popular ones settle on temperature. Energy companies use them as hedges, because a mild winter cuts the volume they sell rather than the price they receive, and a price hedge does nothing about that. Two variables carry most of the market, and for a single day they are defined as
Suppose a day runs between a low of 40 degrees Fahrenheit and a high of 60 degrees Fahrenheit. Then A = (60 + 40)/2 = 50, the daily cooling degree day figure is zero, and the daily heating degree day figure is 15. Contracts settle on the cumulative figure across every day of a stated month, so one freakish afternoon matters much less than the run of the season.
A small set of precious metals is bought as an investment rather than as a raw material: gold and silver above all, with platinum and palladium some way behind. All of them have industrial uses, yet many holders want nothing except exposure to the price, and for those holders a futures or forward contract does the same job as the metal itself.
Two simplifications make this group tractable. Storage costs are small next to the value of the metal, so they can be dropped, and the lease rate earned by lending the metal out is also small. Set both aside and what remains behaves like a financial asset paying no income, so the forward price follows the ordinary relation for such an asset.
Two trades hold that equality in place, and they are worth naming separately because only one of them survives when the commodity is a consumption asset.
Arbitrage A answers a forward price that is too high: borrow S at rate R, buy the metal, and sell it forward for delivery at time T, with the forward obligation covered by the metal already owned.
Arbitrage B answers a forward price that is too low: an owner sells the metal spot for S, invests the proceeds, and buys it back forward for time T. Ownership at time T is unchanged and the difference has been pocketed. That second trade needs a holder content to be without the physical metal for the life of the contract.
Silver trades at USD 26.00 per ounce. The one-year forward price quoted by a dealer is USD 28.00 per ounce, the one-year risk-free rate is 4% with annual compounding, and storage costs are treated as negligible.
The lease rate is what a borrower pays to take an investment commodity out of somebody else’s vault, quoted as a rate on the value borrowed. It plays the part a dividend yield plays for a stock index, except that the owner is paid for lending rather than for holding.
The mechanism is easiest to see through the gold market. A mining company agrees to sell future production forward to an investment bank at a fixed price. The bank is now long gold on a forward basis, so it borrows gold at the lease rate, sells the borrowed metal spot, and puts the proceeds on deposit. Those steps behave like a short forward position and offset the contract written with the producer. Central banks are the main lenders of gold.
With a lease rate of e running alongside the interest rate, the forward relation picks up a second term.
Rearranging gives the lease rate implied by a quoted futures price, which is how the number is usually obtained.
Spot gold stands at USD 1,240. Futures for delivery in six months are quoted at USD 1,250, and the matching risk-free rate, annually compounded, is 4%.
The rate is itself a price, moving with supply and demand for borrowed metal. More hedging by producers means more borrowing by banks and a higher lease rate, less hedging lets it fall, and a greater willingness among owners to lend pushes it down. It turns negative occasionally, and when it does an arbitrageur can buy the metal and sell it forward at a profit.
Now take a commodity held almost entirely for consumption, such as crude oil, copper or corn, and start with no storage costs. If the futures price rose above S(1 + R)T, Arbitrage A would still be open, since buying with borrowed money and selling forward does not require the trader to be a natural owner. That fixes a ceiling.
Once storage enters, the arbitrageur has more to finance. With U for the present value of storage costs over the life of the contract, the borrowing rises to S + U and so does the repayment at maturity.
The lower half of the argument collapses. Arbitrage B asks owners to sell inventory and buy it back forward, and owners of a consumption commodity have plans for that inventory: a refinery to feed, a smelter to run, a mill to keep turning. Selling means shutting something down, so the trade is not attempted and nothing pushes the futures price back up to the bound. Practical frictions block the determined trader too, since the commodity may not be borrowable, storage has to be found and paid for at the going rate rather than the modelled one, and the grade and location on offer rarely match the contract.
The distance between the bound and the traded price is informative, and the convenience yield Y measures it. It answers one question: if this commodity behaved like an investment asset, what yield paid to its holder would explain the quoted futures price?
The benefit is protection: against a shortage, a late delivery, a production line standing idle. When supply is expected to be ample and any quantity can be ordered for near-immediate delivery, the convenience yield sits close to zero and F = (S + U)(1 + R)T is a fair approximation. When inventories are thin, it climbs and the futures price falls well below the bound.
Cost of carry gathers three items into one rate: storage, financing, and whatever the asset pays its holder. For a financial asset the storage term vanishes, leaving a financing rate R against a yield Q.
A currency shows the arithmetic quickly. A domestic interest rate of 4% against a foreign rate of 3% gives carry of roughly 1%. Lift the foreign rate to 6% and carry becomes around -2%, which is negative carry: holding the position costs money over time.
A commodity ordinarily pays its holder nothing, the lease rate on investment commodities being the exception, so carry reduces to interest plus storage. Under continuous compounding the futures price of a consumption commodity can then be written against carry and convenience yield together.
Compounding conventions matter less than they look when rates are small, because a low annually compounded rate is close to its continuously compounded twin. The gap widens as the rate grows, so a large convenience yield has to be converted rather than reused.
Oil trades at USD 69 per barrel, with futures for delivery in six months at USD 65 per barrel. Storing one barrel over that period costs USD 1 in present value terms, against a risk-free rate of 2% a year.
Two words describe the slope of a futures curve, and each is used in two senses. Strictly, normal backwardation puts the futures price under the expected spot price at maturity and contango puts it above. In everyday usage the comparison is made against today’s spot price instead, so a market is in backwardation when deferred contracts trade below spot and in contango when they trade above it.
The cost of carry model explains the everyday version. Contango is carry outweighing the convenience yield, and that is what a carry market means: the curve covers storage and financing, so inventory is held deliberately. Backwardation is the convenience yield dominating, with inventories tight and holders paid to keep the goods.
The slope costs money for anyone holding a commodity through futures rather than in a warehouse, because contracts expire and the position has to be rolled into the next one along the curve. In contango the expiring contract is sold cheap and the replacement bought dear, so the same money buys fewer units, while in backwardation the position gains units. That gain or loss is the roll return, and across a year of monthly rolls it can dwarf the move in the spot price.
A fund holds a commodity through the nearest futures contract and rolls it monthly. Its capital is USD 5,200,000. On the roll date the expiring contract is priced at USD 52.00 per unit and the next contract at USD 53.30 per unit.
The expected future spot price is the market’s average opinion about the price on a given future date. Whether the futures price equals it, and whether it forecasts it well, are separate questions.
Start with the trade that links the two. Let P be the futures price discounted back to today at the risk-free rate, so P = F/(1 + R)T. Investing P now produces exactly F at time T, and a long futures contract turns that F into one unit of the asset. The pair is a synthetic long position in the commodity: it costs P today and delivers ST at maturity, with no storage and no physical delivery to arrange.
Writing X for the expected return on the package, the expected payoff is E(ST) = P(1 + X)T, and substituting for P links the futures price to the expected future spot price.
What decides X is the capital asset pricing model. Systematic risk is the part of an investment’s risk that tracks the market as a whole, taken here to be a broad stock portfolio such as the one behind the S&P 500 Index, and it cannot be diversified away. Nonsystematic risk is the remainder, which a well diversified portfolio removes, so nothing is paid for carrying it. A synthetic commodity position correlated positively with the market earns more than the risk-free rate, one correlated negatively earns less, and an uncorrelated one earns exactly R.
| Correlation with the market return | Systematic risk of the synthetic trade | Expected return X | Futures price against E(ST) | Example |
|---|---|---|---|---|
| Positive | Positive | X greater than R | F below the expected future spot price | Most commodities |
| Zero | None | X equal to R | F equal to the expected future spot price | An asset independent of equities |
| Negative | Negative | X less than R | F above the expected future spot price | Gold |
Source: relationships implied by the capital asset pricing model; the final column is added here.
Most commodities cost more when the economy runs hot, which puts them in the first row. Gold is the standard counterexample, since investors buy it as conditions deteriorate and sell it once equities look attractive again, so its futures price would overstate the expected future spot price. Keynes reached the first row by another route, arguing that speculators must be paid for the risk they absorb while hedgers accept a small average loss for a quieter business, so hedgers short and speculators long puts the futures price below the expected future spot price. Empirically the futures price looks unbiased, yet it is not an accurate forecast.