Forwards · RiskAtoms Academy

Know these first

L1.7.01 What derivatives are and why they exist
L1.3.03 Present value and discounting
L1.6.02 FX forwards and covered interest parity

Math Atoms

∑ Exponents

By the end you will be able to

1Calculate the payoff of a forward and its fair forward price.
2Explain why the forward price is set by the cost of carry, not by a forecast.
3Describe the cash-and-carry trade that keeps forward prices fair.
4Adjust the forward price for income paid by the asset, such as a dividend.

Why it matters

A coffee roaster in Melbourne needs beans in six months. A grower in Brazil wants to know what she will be paid. They can agree the price today with a forward. But what price is fair? It turns out to be a question about interest rates, not about the future of coffee.

Core explanation

Agree today, deliver later

A forward contract is an agreement made today to buy or sell an asset on a fixed future date at a price fixed today. The buyer is long the forward; the seller is short. The price written in the contract is the delivery price, K. Forwards are agreed over the counter, usually between a company and a bank, so the amount, the date and the asset can all be tailored. No money changes hands at the start.

What each side gets. On the delivery date T, the long pays K and receives an asset worth its spot price then, S_T. So the long gains S_T − K. The short gains the opposite. With K = 52.00:

Price at delivery, S_T	Long payoff	Short payoff
45.00	−7.00	+7.00
52.00	0.00	0.00
60.00	+8.00	−8.00

Both sides are obliged to trade, so the payoff is a straight line: every 1 the price rises adds 1 to the long and takes 1 from the short. Many forwards are settled by delivering the asset; others are cash-settled, with the loser simply paying the difference.

What makes a forward price fair?

You might think the fair price is the best guess of the price on the delivery date. It is not. There are two ways to end up owning the asset at T:

Route A: sign a forward

Pay nothing now. Pay the forward price F₀ at T and receive the asset.

Route B: buy now, hold

Borrow S₀ now and buy the asset. At T, repay the loan with interest. This is carrying the asset.

Both routes start with no money out and end with the asset at T. So they must cost the same at T, or someone could buy the cheap route, sell the dear one, and pocket the gap with no risk. That gap-hunting, arbitrage, is what keeps forward prices fair. The fair price is the spot price plus the cost of carry.

Income and storage. If the asset pays income while you hold it, such as a dividend, the holder in route B collects it, so the forward price is lower. If the asset costs money to store, like oil or grain, the forward price is higher. The FX forward of L1.6.02 is the same idea: holding a foreign currency earns the foreign interest rate, which acts like income.

Forward price and value. When a forward is first agreed, K is set equal to the fair forward price, so the contract is worth zero to both sides. Once prices move, it has a positive value to one side and the same negative value to the other.

The formal version

Four symbols: $S_0$ is the spot price today, $S_T$ the spot price at delivery, $K$ the delivery price and $T$ the time to delivery in years. First, the payoff.

Now the fair forward price. Here $r$ is the annual interest rate, compounded once a year, and raising to the power $T$ compounds it over T years (see the Exponents Math Atom).

Adding income. If the asset pays income with present value $I$ before delivery, the holder in route B keeps it, so it comes off the cost:

F_0 = \big(\text{cost of buying now} - \text{value of income received}\big)(1+r)^T

F_0 = (S_0 - I)(1+r)^T, \qquad I = \frac{D}{(1+r)^{t_D}}

where a dividend $D$ is paid at time $t_D$ . Storage costs work the other way: add their present value instead of subtracting.

The formulas, for reference

\text{Long payoff} = S_T - K \qquad \text{Short payoff} = K - S_T

F_0 = S_0(1+r)^T \qquad F_0 = (S_0 - I)(1+r)^T

Textbooks such as Hull use continuous compounding (L1.3.05), which gives the same idea in the form $F_0 = S_0 e^{rT}$ , with r then a continuously compounded rate.

Try it

Start with the share from the worked example: spot 50.00, rate 4%, one year. Then move the dealer's quote above and below the fair price and read the arbitrage. Finally, move the price at expiry and watch the two payoffs mirror each other.

Try itForward lab

Open the futures and forwards model →

1 · Fair forward price

Spot price today, S₀50.00

Interest rate, r (annual)4.00%

Time to delivery, T (years)1.00

Fair forward price F₀ = S₀(1 + r)ᵀ

52.00

2 · A dealer quotes a price

Quoted forward price53.00

Too high: cash and carry. Borrow 50.00 and buy the asset today. Sell it forward at 53.00. At T, deliver it, collect 53.00 and repay 52.00. Riskless profit: 1.00 per unit.

3 · At expiry, with delivery price K = 53.00

long (buyer) short (seller)

Price at expiry, S_T55.00

Long payoff, S_T − K+2.00

Short payoff, K − S_T−2.00

Sum0.00

Straight lines, and no premium: agreed at the fair price, a forward costs nothing to enter.

Worked example

A forward on a European share

A share trades at €50.00 and pays no dividends. The euro interest rate is 4% a year, compounded annually. Try each step before revealing it.

Pricing, payoff and arbitrage

Step 1 of 5

1
Fair 1-year forward price
$F_0 = 50 \times 1.04^1 = 52.00$

Regulatory lens · looking ahead

Forwards are OTC contracts, so each side is exposed to the other failing to pay. Under the BCBS-IOSCO margin framework for non-centrally cleared derivatives, many forwards between financial firms must carry margin, although physically settled FX forwards and swaps are exempt from initial margin. L1.7.11 covers margin; later levels cover the capital banks hold against counterparty credit risk.

Sparks

SparkA forward price is not a forecast

If the forward price of a share is above its spot price, that does not mean the market expects the share to rise. It only means interest rates are positive. Arbitrage pins the forward price to the cost of carry, whatever anyone thinks will happen.

SparkForwards on currencies you cannot deliver

Some currencies cannot be freely moved offshore because of capital controls. Banks still trade forwards on them as non-deliverable forwards (NDFs): at maturity no rupees or won change hands, and the loser pays the difference in US dollars. It is a cash-settled forward.

Key takeaways

A forward fixes today the price K for a trade on a future date T. Both sides must trade.
Payoff to the long: S_T − K. To the short: K − S_T. A forward costs nothing to enter at the fair price.
The fair forward price is the spot price carried forward: F₀ = S₀(1 + r)ᵀ.
If a quote differs from the fair price, cash-and-carry arbitrage locks in the gap.
Income paid by the asset lowers the forward price: F₀ = (S₀ − I)(1 + r)ᵀ. Storage costs raise it.

Reaction

Check what stuck

Question 1 · objective 2

A share with no dividends trades at 80. The interest rate is 5% a year, compounded annually. What is the fair 2-year forward price, to two decimal places?

Question 2 · objective 1

You entered a long forward to buy an asset at 120. At delivery the asset is worth 111. What is your payoff? (Enter a loss as a negative number.)

Question 3 · objective 3

A dealer quotes a forward price above the fair price S₀(1 + r)ᵀ. Which trade locks in a riskless profit?

ABuy the forward and do nothing elseBBorrow money, buy the asset now, and sell it forward at the quoted priceCShort-sell the asset, invest the cash, and buy it forwardDNone: forward prices are forecasts, so no profit is riskless

Question 4 · objective 4

A share trades at 100 and pays a dividend whose present value is 2 before delivery. The rate is 3% a year. What is the fair 1-year forward price, to two decimal places?

Question 5 · objective 2

Why is the forward price of a non-dividend-paying share above its spot price?

ABecause investors expect the share to riseBBecause of the interest cost of buying the share now and holding itCBecause the seller charges a premium for the riskDBecause forward contracts are taxed

Glossary

Forward contract: An agreement today to buy or sell an asset on a fixed future date at a price fixed today.
Long position: The side that agrees to buy. It gains if the price rises.
Short position: The side that agrees to sell. It gains if the price falls.
Delivery price: The price written into a forward contract, paid on the delivery date. Written K.
Forward price: The delivery price that makes a new forward worth zero to both sides today. Written F₀.
Spot price: The price for buying the asset now, for immediate delivery. Written S₀ today and S_T at time T.
Cash and carry: Buying the asset now with borrowed money and holding it until a future date.
Cost of carry: What it costs to hold an asset until delivery: interest, plus storage, minus any income it pays.
Cash settlement: Settling a contract by paying the difference in value rather than delivering the asset.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Determination of Forward and Futures Prices”. Intermediate.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Basel Committee on Banking Supervision and IOSCO, Margin requirements for non-centrally cleared derivatives (BIS, latest revision 2020).

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What derivatives are and why they exist

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Futures and margining