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Lesson 1 of 7 · 11 min

Forward price versus forward value: inception and expiration

The forward price is fixed once at inception so that the contract is worth zero to both sides; at expiration the contract is worth exactly its settlement amount, $S_T - F_0(T)$ to the buyer.

In short

The forward price $F_0(T)$ is agreed at t = 0 and never changes. The forward value $V_t(T)$ is what the contract is worth to one side at time t, and it does change.
At inception the no-arbitrage forward price makes the contract worth zero to both parties: $V_0(T) = 0$ .
With no costs or benefits of holding the underlying, $F_0(T) = S_0(1+r)^T$ : the spot price grown at the risk-free rate.
At expiration the value equals the settlement amount: $V_T(T) = S_T - F_0(T)$ for the long, $F_0(T) - S_T$ for the short.
One side's mark-to-market (MTM) gain is always the other side's MTM loss. Futures settle MTM daily; forwards usually settle only at maturity.

Price and value are different things

Two numbers describe a forward, and the exam keeps them apart. The forward price, $F_0(T)$ , is the price at which the underlying will change hands at maturity T. It is written into the contract at t = 0 and stays fixed for the contract's whole life.

The forward value, $V_t(T)$ , is what the contract is worth to one party at some time t. Think of it as the gain or loss that party would book if the contract were settled right now. This is the contract's mark-to-market (MTM) value.

Because a forward is a zero-sum agreement between two parties, the long's MTM gain is exactly the short's MTM loss, and vice versa. Futures differ only in timing: a clearinghouse settles futures MTM changes in cash every day, while a forward's gain or loss is usually settled once, at maturity.

	Forward price $F_0(T)$	Forward value $V_t(T)$
Set when?	Once, at t = 0	Recomputed at any t
Changes over the life?	No	Yes: spot, time, rates
At inception	No-arbitrage level	Zero
At maturity	Still $F_0(T)$	$S_T - F_0(T)$ for the long

At inception: value is zero

Entering a forward costs nothing in cash (ignoring transaction costs and counterparty credit risk). For that to be fair, the forward price must be set so that neither side is getting an asset or taking on a liability. That makes $V_0(T) = 0$ for both the buyer and the seller.

The price that achieves this is the no-arbitrage price from the replication argument. Buying the asset today and holding it to T costs $S_0$ plus the opportunity cost of the cash tied up, measured by the risk-free rate. With no other costs or benefits, $F_0(T) = S_0(1+r)^T$ .

For example, a share in a Norwegian company trades at NOK 250, the one-year risk-free rate is 3% and the share pays no dividend. The one-year forward price is $250 \times 1.03$ = NOK 257.50. At that price, a forward buyer and seller both hold a contract worth zero on day one.

Zero value at inception does not mean the forward price equals the spot price. The forward price is above spot here because whoever buys forward keeps their cash invested at r until T.

At expiration: value equals the settlement amount

At maturity the long pays $F_0(T)$ and receives an asset worth $S_T$ . Its value is therefore $V_T(T) = S_T - F_0(T)$ . The short delivers an asset worth $S_T$ and receives $F_0(T)$ , so its value is $F_0(T) - S_T$ .

These payoffs are symmetric: every unit the spot price ends above the forward price is a unit gained by the long and lost by the short. In the NOK example, if the share ends at NOK 270, the long gains 270 − 257.50 = NOK 12.50 per share and the short loses the same. If it ends at NOK 240, the long loses NOK 17.50 and the short gains NOK 17.50.

Whether the contract is settled by physical delivery or in cash, the economic result is the same: the settlement amount is the value at maturity.

Forward price NOK 257.50. The long and short payoffs are mirror images that cross zero exactly at the forward price: a gain for one side is always the same-sized loss for the other.

Formulas

Value at inception: $V_0(T) = 0$
Holds for both parties when the forward price is set at its no-arbitrage level (ignoring transaction costs and credit risk).
Forward price, no costs or benefits: $F_0(T) = S_0(1+r)^T$
S₀ = spot price at inception, r = annual risk-free rate, T = years to maturity.
Value at maturity (long): $V_T(T) = S_T - F_0(T)$
For the short, reverse the sign: F₀(T) − S_T.

Worked examples

Example 1 Settlement value for both sides

A fund agrees to buy a non-dividend-paying share forward at NOK 257.50 in one year. What is the contract worth to the fund (long) and to the dealer (short) at maturity if the share ends at NOK 270? At NOK 240?

Example 2 Pricing and settling a six-month forward

A share trades at JPY 4,000 and pays no dividend. The risk-free rate is 1.5%. (1) What is the six-month forward price? (2) An investor sells 500 shares forward at that price. What is the dealer's (long) settlement value in total if the share ends at JPY 3,950?

Example 3 Price or value?

Three months into a one-year forward, the spot price has risen sharply. A colleague says the forward price has gone up. Is that right?

On the exam

Expect items asking for the value at inception (zero), the value at maturity from the long's or short's side, or matching situations with 'buyer gains', 'seller gains' or 'value is zero'.

Saying the forward price changes during the contract's life. It is fixed; only the value changes.
Thinking the forward price must equal the spot price because the value at inception is zero.
Zero value at inception is the rule for a forward struck at its no-arbitrage price, and that is what conceptual items expect. A contract deliberately struck away from it (off-market) is worth $S_0 - F_0(T)(1+r)^{-T}$ to the long from day one.
Getting the sign wrong for the short: its value at maturity is $F_0(T) - S_T$ .

Check yourself

Q1A share trades at CHF 60, pays no dividend, and the one-year risk-free rate is 2%. The no-arbitrage one-year forward price is closest to:
Q2At inception, a forward priced at its no-arbitrage level has a value to the buyer that is:
Q3A party is short a forward at AUD 45.00. At maturity the underlying trades at AUD 48.30. The value of the contract to the short is:
Q4Which statement about a forward contract during its life is most accurate?
Q5At maturity, the spot price is exactly equal to the forward price. The value of the contract is:

0/5Check yourself

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