Futures Fair Value, Explained (With a Calculator)

Spot, carry, and the theoretical futures price — why F = S·e^(r−q)T, what enforces it, and how to use a futures fair value calculator to check any contract.

Futures fair value is the theoretical price at which a futures contract should trade given the current spot price, the cost of financing a position, and any income or costs the underlying generates between now and expiration. It is not a prediction of where the underlying will be at delivery — it is a present-value relationship enforced by arbitrage. When you see a financial news anchor say "fair value on the S&P 500 futures is −12 points" before the open, they are reporting the exact same number: the differential between the index level and the theoretical futures price implied by carrying the index overnight. Use the futures pricing calculator to compute fair value for any contract in seconds.

The cost-of-carry model

The workhorse formula is the cost of carry model. For a continuously compounded rate and a continuous yield, fair value is:

F = S · e^{(r − q)T}

where:

• S is the current spot price of the underlying.
• r is the continuously compounded risk-free rate (financing cost).
• q is the continuous yield on the underlying — dividend yield for an equity index, foreign risk-free rate for FX, or net convenience yield minus storage cost for a commodity.
• T is time to expiration in years.

If you prefer discrete compounding, the equivalent expression is F = S · (1 + r)^T − PV(income), where PV(income) is the present value of all distributions paid by the underlying before expiry. Both forms say the same thing: to hold the underlying through expiration, you pay financing and receive yield; the futures price adjusts so neither trade is free money.

What r and q represent for different underlyings

The symbols r and q absorb very different economics depending on the asset class:

• Equity index futures (e.g., S&P 500, Nasdaq-100). r is the short-term funding rate (historically T-bill yield or SOFR), and q is the dividend yield on the index. Because large-cap indices pay dividends, q lowers the futures price relative to spot — the index holder receives dividends that the futures holder does not. When dividend yield exceeds the funding rate, futures trade at a discount to spot.
• FX futures (e.g., EUR/USD). By covered interest parity, r is the domestic rate and q is the foreign risk-free rate. The formula becomes the standard interest-rate parity equation; a currency with a higher rate trades at a forward discount.
• Commodity futures (e.g., crude oil, gold, corn). q becomes the net convenience yield — the implicit value of holding physical inventory, such as the ability to keep a refinery running — minus the storage cost. When storage costs dominate, futures trade above spot (contango). When convenience yield dominates, futures trade below spot (backwardation).

The no-arbitrage argument that enforces fair value

Fair value is not merely theoretical — it is patrolled by two opposing arbitrage strategies that collapse any deviation back to formula. This is the logic behind spot-futures parity.

Cash-and-carry (futures too expensive). If F > S·e^(r−q)T, an arbitrageur borrows S dollars at rate r, buys the underlying, and shorts the futures at F. At expiration they deliver the underlying and collect F, repay the loan at S·e^rT, and pocket the dividends worth S·e^qT along the way. Net profit: F − S·e^(r−q)T > 0, risk-free. Selling pressure on the futures and buying pressure on spot close the gap.

Reverse cash-and-carry (futures too cheap). If F < S·e^(r−q)T, an arbitrageur shorts the underlying (or sells out of a long position), lends the proceeds at r, and goes long the futures at F. At delivery they take the futures delivery and reconstruct the position. Net profit: S·e^(r−q)T − F > 0, risk-free. Buying pressure on futures and selling pressure on spot again closes the gap.

In liquid markets with low transaction costs — large equity-index futures being the canonical example — these two bounds are tight enough that deviations rarely exceed a few basis points of notional for more than seconds.

Worked example: S&P 500 index future

Suppose the following inputs at 9:29 a.m. Eastern on a Monday:

Input	Value
S&P 500 spot (SPX)	5 480.00
Days to front-month expiration	18 days (T = 18/365 = 0.0493)
3-month SOFR (annualized)	4.80% continuously compounded
SPX dividend yield (annualized)	1.30% continuously compounded

Applying the formula: F = 5 480 · e^{(0.0480 − 0.0130) × 0.0493} = 5 480 · e^0.001726 ≈ 5 480 · 1.001727 ≈ 5 489.46

So fair value for the E-mini S&P futures contract (ES) is approximately 5 489.46. If the ES is trading at 5 491.00, it is roughly 1.54 points rich to fair value — a modest premium that would narrow heading into expiration.

The "fair value" figure broadcast on financial TV before the open is essentially this number inverted: given where futures are trading overnight, by how many points does the cash index need to open to be in equilibrium? A futures market showing ES at 5 494 against a fair value of 5 489 implies the cash market should open roughly 4–5 points above the prior close.

The basis and why futures rarely sit exactly at fair value

The basis is the difference between the futures price and spot: basis = F − S. In an efficient market the basis should equal the carry cost S·(e^(r−q)T − 1). In practice a few persistent forces create small, temporary deviations:

• Dividend uncertainty. Future dividends on an index are not perfectly predictable. If the market revises its dividend estimate, fair value shifts without any change in spot or rates.
• Financing rate moves. Intraday changes in repo or SOFR alter r in real time.
• Supply and demand imbalances. Large institutional hedging flows — pension funds rolling index futures, for instance — can push futures temporarily away from fair value before arbitrageurs step in.
• Transaction costs and margin. Arbitrage is only profitable if the deviation exceeds the round-trip cost of cash-and-carry. The bandwidth inside which no arbitrage is triggered is called the no-arbitrage band.

For liquid equity-index futures that no-arbitrage band is narrow — typically just a few index points on a contract worth hundreds of thousands of dollars. For less liquid commodity futures or longer-dated contracts, the band can be meaningfully wider because storage and convenience yields are harder to observe and hedge precisely. Plugging your inputs into a futures pricing calculator gives you the theoretical center of that band immediately.