2. The price of a future is not a forecast: cost of carry and basis

Derivatives 1: Futures

This module corrects the most widespread misunderstanding about derivatives, and it is worth saying it bluntly from the first line: the price of a future is not what the market thinks the underlying will be worth on the expiry date. Almost everybody has it the wrong way round, and whole investment decisions are born of that mistake.

A future is not an opinion poll

When you read that the six-month oil future is trading at 84 dollars while the barrel today is at 80, the intuitive conclusion is that the market expects oil to rise by four dollars. It is false. That gap does not come from any expectation: it comes from an arithmetical calculation of what it costs to keep the barrel in storage until then.

If you want to own a barrel six months from now you have two routes. One: buy it today at 80 and store it. Two: buy a future and pay in six months' time. If both lead to the same place, their costs have to match, because otherwise there would be free money sitting on the table. And the first route has very concrete costs.

Cost of carry: funding, storage and whatever the asset yields

That set of costs is called the cost of carry, and it has three components:

• Funding. Buying today ties up money. Either you borrow it and pay interest, or it is your own and you give up what it would have earned you risk-free. In both cases there is a real cost, the same cost of money you already know from the macro course.
• Storage and its trimmings. If the underlying is physical, it has to be kept, insured and allowed to lose a little along the way. A crude oil tank is not free; a stock index, by contrast, takes up no space.
• Whatever the asset yields in the meantime, which enters as a subtraction. If you hold the shares in an index, you collect their dividends, and that income offsets part of the funding.

Hence the idea worth memorizing: the price of the future equals the spot price plus what it costs to carry the asset to expiry, minus what that asset pays you along the way. None of this contains an opinion about the future.

A three-month calculation, step by step

Let us make it concrete. A share is trading at 100 euros, the interest rate is equivalent to 4% a year and the company pays dividends at a rate equivalent to 2% a year.

• Three months of funding: 4% a year over a quarter of a year is 1%, that is, 1 euro.
• Three months of dividends: 2% a year over a quarter of a year is 0.5%, that is, 0.50 euros, which subtract.
• Net cost of carry: 1 minus 0.50, that is, 0.50 euros.

The three-month future ought to trade at around 100.50 euros. Notice what has not appeared: no forecast, no analyst, no earnings estimate. Just an interest rate and a dividend.

And note the uncomfortable consequence: if the expected dividend were 6% a year, the net cost would be negative and the future would trade below spot, at around 99.50 euros. A future below today's price does not mean falls are expected. It may simply mean the asset pays a lot.

The basis and its convergence

The difference between the spot price and the future's price has a name of its own: the basis. It is usually defined as spot minus future, though you will find the opposite convention depending on the market. What matters is not the sign but this property: the basis tends to zero as expiry approaches.

The reason is almost tautological. The cost of carry is paid for a length of time, and that time gets shorter day by day. Three months from expiry there are three months of funding to cover; the next day, one less. When expiry arrives there is nothing left to fund and both prices describe the same thing: buying the asset now. At that moment they coincide. That forced convergence is what makes a future usable for hedging risk.

The arbitrage that holds it all in place

Who forces the price to behave this way? Nobody in particular, and that is why it works. If the future strays from its theoretical value by enough to cover commissions, spread and funding, a riskless trade appears: you buy the asset spot with borrowed money, simultaneously sell the future and wait until expiry to deliver. The result is locked in from the first minute.

If the deviation runs the other way, the trade is done in reverse: you sell the asset spot, place the money at the interest rate and buy the future. In both cases, the trade itself pushes the prices toward each other and the deviation closes on its own. From which something important follows: arbitrage keeps the price in place for as long as it can be executed.

A real case: March 2020, when gold arbitrage seized up

At the end of March 2020, with markets in full pandemic panic, something happened that in theory can hardly happen at all: the gold future in New York parted company with the spot price in London by far more than any cost of carry could justify. A gap that in normal conditions is counted in cents or a few dollars an ounce reached several tens of dollars.

On paper there was free money: buy gold in London, sell the future in New York and deliver. In practice, no. Commercial flights, which carry a good part of the metal, were suspended. The Swiss refineries that recast bars into the format required by the New York contract had stopped because of lockdown. And anybody wanting to set up the trade needed margin and balance sheet just when both were scarcest.

The lesson is not that the theory failed, but that the theoretical price rests on a silent assumption: that somebody can execute the arbitrage. When that assumption breaks, the relationship between spot and future breaks with it, and it does so precisely on the days when it is most needed to hold.