Forward Price

Builds onFuture Value of a Single Sum — if this page feels steep, start there.

F_0 = S_0 (1 + r)^T

$F_0$: the forward price: agreed today, paid at delivery
$S_0$: the spot price: what the asset costs for immediate delivery today
$r$: the risk-free rate — the cost of financing the carry trade
$T$: time to delivery, in years
$(1+r)^T$: the financing growth factor: what the borrowed purchase costs by delivery

Why it must be true

What should you agree today to pay for an asset delivered in a year? Not a forecast — a replication. The seller can buy the asset right now for $S_0$ with borrowed money and simply hold it until delivery. By then the loan has grown to $S_0(1+r)^T$ . If the forward price were higher, sellers would mint riskless profit doing exactly that; if lower, the reverse trade wins. Competition pins the forward at the cost of the carry trade — the spot price, future-valued.

The shocking part for newcomers: the expected future price appears nowhere. Forwards are priced by arbitrage, not by opinion.

The derivation

Build the delivery two ways and demand they cost the same.

Way 1: agree the forward — pay $F_0$ at time $T$ , receive the asset.

Way 2: borrow $S_0$ , buy the asset now, hold it; at $T$ repay $S_0(1+r)^T$ and you hold the same asset.

Identical outcomes must have identical prices:

F_0 = S_0(1+r)^T

Assets that pay income while held (dividends, coupons) reduce the carry cost — subtract the income's PV from $S_0$ before compounding; storage costs add. The general recipe: future-value the NET cost of carrying.

When to reach for it

Pricing a forward or futures contract on an asset given its spot price and the risk-free rate — the no-arbitrage carry argument.

Listen for

“forward price / futures price”“no-arbitrage price for delivery in …”“spot price of … risk-free rate of …”“cost of carry”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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One year, small rate: F ≈ S(1+r) — a 5% rate lifts a $100 spot to$ 105 forward. The forward premium over spot IS the interest rate.
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The forward must sit ABOVE spot for a no-income asset with positive rates. A candidate answer below spot is the discounting-direction error.
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Expected-price talk in a question is a decoy — arbitrage pricing never uses it.

Traps in applying it

✗Discounting the spot instead of compounding it — the forward is a FUTURE value.
✗Ignoring dividends or storage: income while holding lowers the forward; costs raise it. The bare formula is for a no-income asset.
✗Mixing the rate and period units (a 6-month forward wants (1+r)^0.5 or a semiannual rate, not a full year).

Limits & criticisms

The clean arbitrage needs frictionless shorting, borrowing at the risk-free rate, and costless storage — real markets charge for all three, so actual forwards live inside a band around the formula rather than on it. Convenience yields on commodities (the value of physically holding oil when supplies are tight) can push futures below the carry price entirely, producing backwardation the bare formula can't explain.

Where it came from

Forward contracts are ancient — Mesopotamian grain deals and Osaka's Dōjima rice market (1730s) traded them — but the cost-of-carry arbitrage pricing was formalized with the growth of modern futures markets after the CME's financial futures launched in 1972, and became textbook canon through the derivatives revolution of the 1980s. Every futures desk, FX forward quote and commodity curve is this logic; when markets violate it (as in the 2020 negative-oil episode, when storage broke), the exceptions prove the mechanics.

One identity, 1 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

The carry-trade price

F_0 = S_0(1+r)^T

The no-arbitrage face: the forward is the spot future-valued — priced by replication, never by forecast.

Drill this face →

On the BA II Plus

Worked example: With spot at $135.00 and the risk-free rate at 2.5%, what forward price for delivery in 1.5 year(s) admits no arbitrage?

1.1 [+] 0.025 [=] [y^x] 1.5 [=]the financing factor (1+r)^T
2.[×] 135 [=]times spot = forward price

→ $140.09

Where it leads

Master this and the following come almost for free:

Put–Call Parity One-Period Binomial Option Value

Drill this formula Back to the library