The Futures Price Formula for Commodities Requires Additional Ingredients

Such commodities can be predominantly investment-oriented (e.g., gold, silver) or predominantly consumption-oriented (e.g., copper, oil). The latter requires taking additional factors into account.

The basic formula for calculating forward/futures prices (for a non-dividend-paying stock) is:

(more detail: Valuing Forward Contracts)

Now let’s see what form this formula takes for investment and consumption commodities.

Formula for Investment Commodities

Unlike financial securities, commodities involve physical storage costs. At the same time, investment commodities typically also generate income (for example, gold can be lent by central banks, earning interest income).

If we denote the net present value of storage costs and income by U, then the basic formula becomes:

If, for practical reasons, we assume that the storage costs and benefits are proportional to the price of the commodity, then the formula takes this form (where u is the percentage rate):

Formula for Consumption Commodities

In the case of consumption commodities, the above proofs do not hold. If we return to the initial logic, the proof starts with inequalities and showing that arbitrage forces the inequality to become equality.

If not, then a trader could earn a risk-free profit with the following combination:

  • Borrowing money at the risk-free rate and buying the commodity.
  • Selling the commodity forward (short position in a forward contract).

Thus, because of arbitrage, such an inequality cannot persist for long…

But what happens when the situation is the reverse?

In the case of investment commodities, we would say that risk-free profit is still possible:

  • Sell the commodity, save on storage costs, and invest the money at the risk-free rate.
  • Enter a long futures contract to buy the commodity later.

However, this argument does not work for consumption commodities. Owners cannot easily sell the commodity and replace it with a futures contract in production. Thus, the arbitrage argument is not sufficient to turn inequality into equality, and in the end, for consumption commodities we obtain the following inequality:

Futures price is less than or equal to the spot price plus the future value of net storage costs.

Or, if the costs are proportional to the price, then the relationship looks like this (with percentages instead of absolute values):

Convenience Yield

Physically holding consumption commodities has its own benefit, called the convenience yield.

Such commodities allow producers to continue production without interruption and earn additional profits when supply is temporarily limited. The convenience yield is calculated with this formula:

If we assume storage costs are proportional to the price, then the formula transforms, and in the end, we get:

In essence, the convenience yield determines how much lower the futures price will be relative to the future value of the current spot price. It defines the size of the gap in this inequality:

P.S.

In the table below, it can be seen that the futures price of soybeans decreases as the contract maturity increases. This indicates that the benefit from holding soybeans is high; in other words: [y > r + u]

Sometimes ( r + u ) is written as c – the cost of carry.

  • For investment commodities, we get:
  • For consumption commodities, we get:

Where, if ( c > y ), then the futures price is an increasing function of contract maturity; and conversely, if ( c < y ), then increasing the maturity leads to a decrease in the futures price.

Source: Options, Futures & Other Derivatives, John C. Hull