Futures Pricing

In the previous section, we established the relationship between spot (S) and futures (F) prices as S = F d(0,T), where d(0,T) is the discount rate over the period of the futures contract. If it is observed that S < F d(0,T) for which the current spot price is below the present value of the futures price there will be an incentive to short the futures contract, buy and store the asset, and then deliver it at time T for F. If, on the other hand, S > F d(0,T), there is an incentive to short the asset, invest the proceeds at d(0,T) and go long the futures, taking delivery of the asset at time T for a price F. Either way, there is an arbitrage profit to be made, and since these opportunities will be driven from the market, we conclude that the relationship between spot and futures prices must be fully consistent with the existing term structure of interest rates. Let's now take a closer look at this logic.

Suppose you borrow an amount S, buy one unit of the underlying asset on the spot market at price S, and take a short position in the futures market (to deliver this asset at price F). The total cost of this portfolio is zero, that is, the portfolio is [–S/d(0,T), F]. At time T, you deliver the asset for F and repay the loan amount S/d(0,T). This is an equilibrium result. This relationship says that the current spot price is equal to the discounted present value of the futures price. If there were storage costs involved, then F must compensate for those as well as described earlier. Whereas the basic relationship is:

we now have with storage costs:

The summation is over the k times in which the cash flows (storage cost payments) occur. Notice that storage costs are compounded, meaning that the futures price must necessarily be higher to compensate for these additional costs. Stated equivalently, the spot price must equal the discounted present value of the future price minus the compounded storage costs. We can solve this relation for F:

Futures Returns and the Futures Term Structure

We dealt with a few cases earlier that computed the return to the futures contract. In general, we showed that the effective price paid by the hedger was the sum of the realized spot price plus the return to the futures contract: S₂ + (F₁ – F₂). Let's focus on the latter term. The hedger shorts (sells) the futures contract at time t = 1 for F₁ and when he closes out his position at t = 2, he realizes the gain F₁ – F₂. As a short seller, a profit is earned when the futures price declines. As a hedger taking a long position, a profit is earned if F₂ > F₁. The profit or loss in either case is the result of closing out the contract.

In practice, futures positions are often rolled forward, especially in cases in which the hedger wishes to maintain a futures exposure over a period of time. Rolling the futures contract means selling the current contract before expiration and replacing that contract with the next available contract. Table 15.2 illustrates how futures are rolled forward. So, for example, if we desire to maintain exposure to oil futures through long one-month futures contracts then we must sell the current contract before it expires (to avoid taking delivery) and replacing it with the next available one-month contract. Using the following table, let's assume that contracts are rolled on the twenty-fourth day of the month. To maintain front-month exposure then, the current month contract is sold and the proceeds (and any additional funds if necessary) are used to buy the next month's contract.

Table 15.3 Rolling Futures Contracts.

The December contract in the table is therefore sold for $52.91 on November 24th. This is the analog to F₂ from before, and if F₂ > F₁, then a profit is recorded on the futures position when closed. We maintain our front month exposure by buying the January contract for $61.68. When the contract is cleared the following day, we record a roll yield of –14 percent. We do the same on the 24th of December for a roll yield of 3.25 percent by selling our January contract for $76.87 and buying the February contract for $75.08.

You will undoubtedly notice that the roll yield depends on the relationship between futures prices across contracts—the so-called term structure. The futures term structure can be seen by looking across a given row; we can see that it is humped in the preceding table, rising from the December to the January contract and then falling off when rolling to the February contract. An upward-sloping futures term structure is referred to as contango, while a downward-sloping term structure is called backwardation. (Contango is said to be a corruption of the English word continue and was thought to arise from the practice of buyers of commodities asking sellers for a continuation in the contract date. The sellers would do so only if they were compensated for the time value, or opportunity cost, of forgoing payment for the continuation period. Thus, longer continuations [contracts further out] sold at a premium.) Erb and Harvey (2005) and Gorton and Rouwenhorst (2004) argue that the realized historical positive returns to long futures exposures to commodities are explained in large part to backwardation and resultant positive roll yields and not to growth in commodity prices. Contango, on the other hand, produces negative roll yields that can quickly wipe out any positive returns from the futures position itself and, as a result, we are naturally interested in which factors affect the shape of the futures term structure.

Strictly speaking, if futures prices exceed expected spot prices, for example, F(t) > E[S(t)], then the futures market is said to be in contango and roll yields will be negative. Contango is often identified if F(t + 1) > F(t), ostensibly due to the belief that the expected spot price is not observed. For this to occur, the futures market has to have an imbalance of hedgers who are naturally short the commodity relative to hedgers with long positions. Speculators who are willing to take long positions will enter into contracts with short hedgers only if they are adequately compensated. For example, the futures prices must compensate them for storage costs and the risk that prices may fall. Contango is normal for nonperishable commodities with costs of carry (storage costs). In this situation, the longer-term contracts sell at a premium to nearer-term contracts due to the costs of carry. Theoretically, the size of the contango should be limited by the costs of carry but that has not always been the case, especially when the imbalance between short hedgers and long speculators is high and spot prices are volatile and uncertain.

On the other hand, when the prompt (nearest-term) contract sells at a premium to contracts further out, the market is said to be in backwardation. Backwardation in futures is normal in markets dominated by long hedgers; hedgers and speculators can be induced to take the short position (taking delivery) if the futures price is below the expected spot price. The longer the time to delivery, the greater the spread F(t) < E[S(t)] necessary to protect against possible increases in spot prices. On the other hand, expectations of impending shortages can cause the holder of a commodity to charge a convenience yield that reflects the benefit of having the commodity in hand now than having to wait for it later. Similar thinking explains backwardated markets for commodities that pay dividends. Therefore, backwardated commodity markets have a declining term structure (see the December 24th January to February roll discussed earlier).

In 2008, crude oil futures were backwardated until early summer when they suddenly shifted into contango. This could happen as traders bought oil on the spot market and sold it forward. They would store the oil until delivery and the costs of carry contributed to the contango. The contango built into the market for commodities has exceeded the costs of carry in general; that is, future prices have exceeded the costs of owning physical commodities, which helps explain the increased interest from traders in owning inventories or warehousing and storage facilities. This behavior, in turn, has incented overproduction of commodities through the recession that began in 2008 while having only minimal impact on declining prices. But, as storage availability declined, making it more difficult to absorb and store the excess supply, the contango eventually eroded until it was equal to the actual costs of finance, returning contango to the cost of carry upper bound.

Agents who had access to lower financing costs could effectively exploit contango opportunities, creating a cash and carry trade that allowed them to sell futures with much lower storage costs. Institutional investors, like pension funds, with their interests in commodity investments as a natural inflation hedge, acted as natural counterparties to these trades by establishing the required long positions. Because they rolled their positions regularly they were the perfect counterparty to the cash and carry trade characterized by short futures positions with no intention of making delivery.

Since commodity prices are generally positively correlated with inflation, they then become natural candidates as inflation hedges (although there are probably more direct hedges such as TIPS and CPI swaps). There has been an active interest in commodities as portfolio diversifiers as well as hedging instruments but controversy still exists as to the effectiveness of these strategies. Some of that controversy is linked to the fact that outside of supply and demand, commodities do not possess a set of underlying fundamentals (for example, dividends and earnings) on which to base trading strategies.