Cost Of Carry Model

Ask someone what a futures price means and most will say "the market's guess of where the asset will be." That is wrong, and it is wrong in a useful way. A futures price is not a prediction. It is an arithmetic fact about two ways to own the same thing on the same future date, and if the two ways ever priced differently, you could lock in free money today.

Two ways to own gold in three months

Suppose you want an ounce of gold delivered in three months. Route one: buy it today, pay cash, and store it in a vault for three months. Route two: sign a futures contract now and pay for it in three months. Both routes end with you holding the same ounce of gold on the same day, so they must cost the same in today's dollars — otherwise everyone would take the cheap route and sell the expensive one, pocketing the gap for free. Route one costs you the price of gold today, plus the interest you gave up by not investing that cash elsewhere, plus the vault's storage fee, minus nothing (gold pays no dividend). That total is what the futures price must equal. This total toll — financing plus storage minus any income the asset throws off — is the cost of carry.

The formula

$$F_0 = S_0 \, e^{(r + u - y)T}$$

In plain English: the futures price $F_0$ equals the spot price $S_0$ grown at a rate that bundles three things — $r$, the risk-free interest rate you forgo by tying up cash; $u$, the storage cost as a percentage of the asset's value per year; and $y$, the yield the asset pays you while you hold it (dividends, coupons, or a commodity's convenience yield), which you don't get if you wait and buy later. $T$ is the time to delivery in years. For a stock index paying a steady dividend yield, $u = 0$, and the formula shrinks to $F_0 = S_0 e^{(r-y)T}$.

The discrete-compounding version, easier for hand arithmetic: $F_0 = S_0(1+r)^T - \text{FV(dividends)}$, where you grow the spot price by the interest rate and subtract the future value of any cash the asset paid out before delivery.

Worked example 1: a stock index future

The S&P 500 index is at 5,000. The risk-free rate is 5% a year, the index's dividend yield is 1.5% a year, and the future expires in 3 months ($T = 0.25$). No storage cost applies to an index.

$$F_0 = 5{,}000 \times e^{(0.05 - 0.015)(0.25)} = 5{,}000 \times e^{0.00875} = 5{,}000 \times 1.00879 = 5{,}044$$

The fair futures price is about 5,044 index points — 44 points above spot. That premium is pure arithmetic: it is the interest earned on $5,000 for three months (about $62.50 worth) minus the dividends you'd have collected by owning the index directly (about $18.75 worth), converted back into index points. Nobody predicted the market would rise; the futures price is higher purely because holding cash and buying later beats holding stock and collecting dividends, by exactly that margin.

Worked example 2: a commodity in contango

Crude oil spot is $70 a barrel. Financing costs 5% a year, storage (tank rental, insurance) costs 3% a year, and oil has no dividend, so $y = 0$. For a 6-month future ($T=0.5$):

$$F_0 = 70 \times e^{(0.05 + 0.03)(0.5)} = 70 \times e^{0.04} = 70 \times 1.0408 = 72.86$$

The future trades at $72.86, about $2.86 above spot. If it traded above that — say $75 — you could borrow money, buy oil at $70, pay to store it, and simultaneously sell the future at $75, locking in a profit with no market view whatsoever. That trade, cash-and-carry arbitrage, is what keeps the market at $72.86 and nowhere else.

Watch the sign of the carry

Even though this explorer is built for interest-rate curves, the same shape logic applies to futures curves: when carry is positive ($r + u > y$), each further-dated future costs more than the last — contango. When the yield or convenience yield dominates, later futures are cheaper — backwardation. Watch how a curve's slope flips as the balance of "cost to hold" versus "benefit of holding" changes; see Contango and Backwardation for the market-structure consequences.

What this means in practice

Cost of carry is the backbone of every relative-value trade between spot and futures markets, and every calendar spread between two futures maturities: the fair difference between two futures on the same underlying is just carry over the gap between their expiries. Index arbitrage desks, ETF creation/redemption units, and commodity storage traders all monitor the gap between the model price and the traded price, because that gap is the arbitrage.