CFA Level I Exam · Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Pricing and Valuation of Forward Contracts: Forward Price vs Forward Value

Updated 7 October 2026 · Fact-checked

The forward price is the delivery price fixed at initiation so the contract starts with zero value: F0 = S0 × (1 + r)^T, adjusted for income and carry costs. The forward value later is the long's gain: adjusted spot minus the present value of the original forward price. At expiration, value = ST − F0.

Understand Pricing and Valuation of Forward Contracts

A forward contract is an agreement to buy or sell an asset at a fixed price on a future date. Nobody pays anything at the start, so the contract must be worth zero to both sides on day one. The fixed price that makes this true is the forward price, F0.

Why that price? Use arbitrage and replication. You can buy the asset today with borrowed money and hold it until T. You then owe S0 grown at the risk-free rate. If the forward price were higher, you would borrow, buy the asset and sell it forward to lock in a risk-free profit. If it were lower, you would short the asset, lend the proceeds and buy forward. The no-arbitrage price is the cost of carrying the asset to T.

Holding the asset can bring benefits (dividends, interest, a convenience yield) and costs (storage, insurance). Benefits lower the forward price because the forward holder does not receive them. Costs raise it. This is the cost of carry idea.

After initiation, the forward price for a new contract changes as spot changes, but your old contract is locked at F0. So you must separate two things. The forward price is the agreed delivery price and never changes for that contract. The forward value is what the contract is worth now. It starts at zero and moves with the spot price and time. At expiration the long's value is simply ST − F0.

Key formulas to remember

Forward price, no cash flows (discrete compounding): F0 = S0 × (1 + r)^T; r is the annual risk-free rate, T is in years. Reasoning: the cost of buying the asset and financing it to T.
Forward price with income and carry costs: F0 = (S0 − PV(benefits) + PV(costs)) × (1 + r)^T; Discount each benefit or cost from its payment date back to today at the risk-free rate before adjusting spot.
Forward price, continuous compounding: F0 = S0 × e^(rT); With a continuous yield q on the asset: F0 = S0 × e^((r − q)T). With a continuous carry cost θ, use r + θ − q.
Value of a long forward at time t (no cash flows): Vt = St − F0 ÷ (1 + r)^(T − t); Value of the short is −Vt. Continuous version: Vt = St − F0 × e^(−r(T − t)).
Value of a long forward at time t (with income or costs): Vt = (St − PV(benefits) + PV(costs)) − F0 ÷ (1 + r)^(T − t); PV here covers only the benefits and costs still to come between t and T.
Value at expiration: VT = ST − F0; For the long. The short's value is F0 − ST.
Value at initiation: V0 = 0; This is why the forward price is set as it is.

How to solve Pricing and Valuation of Forward Contracts questions

Decide first whether the question asks for a forward price or a forward value, and whether it is at initiation, during the life or at expiration. Then follow these steps.

1Identify what is asked: new forward price F0, value of an existing forward Vt, or expiration value VT. Note whether you are the long or the short.
2List the inputs: spot price (S0 or St), risk-free rate r, time to expiry (T for price at initiation, T − t for value later), and any income or costs with their dates.
3Check the compounding convention. Use (1 + r)^T for discrete rates and e^(rT) when the rate is continuously compounded.
4Present value any income or carry costs to the relevant date using the risk-free rate. Subtract benefits from spot and add costs.
5For a price at initiation, grow the adjusted spot to T: F0 = (adjusted S0) × (1 + r)^T.
6For a value during the life, compute adjusted St, then subtract F0 discounted over the remaining time T − t.
7For a short position, flip the sign of the long's value.
8Sanity check: value at initiation should be zero, and a higher spot should raise the long's value.

Quickest way: Spot minus discounted forward price

When to use it: Use it for any value-during-life question. Under time pressure it saves you from computing a new forward price.

Adjust the spot: subtract PV of remaining benefits, add PV of remaining costs. With none, use St.
Discount the original F0 back over T − t: F0 ÷ (1 + r)^(T − t).
Subtract: Vt = adjusted spot − discounted F0. Positive means gain for the long.
On the BA II Plus, key 1.04 [yx] 0.75 [=] to get (1.04)^0.75. On the HP 12C, key 1.04 [ENTER] 0.75 [yx]. For e^x, enter x first, then press [2nd][ex] on the BA II Plus or [g][e^x] on the HP 12C.
Eliminate options: if an option equals St − F0 with no discounting, it is usually the trap. When r is positive, the undiscounted St − F0 understates the long's value, so the correct value is larger than St − F0.

Common mistakes in Pricing and Valuation of Forward Contracts

Treating the forward price and the forward value as the same thing.
Both are called 'forward' numbers and both are quoted in currency units.
Fix: The forward price F0 is the fixed delivery price. The forward value Vt is the contract's current worth and is zero at initiation. Ask what the question wants before you calculate.
Using Vt = St − F0 before expiration.
It is the correct formula at expiration, so students apply it all the time.
Fix: Before expiry, discount F0 over T − t at the risk-free rate. Only at T does the discounting vanish.
Subtracting dividends without discounting them, or discounting them for the wrong period.
Students treat the income as a plain adjustment to spot.
Fix: Discount each cash flow from its payment date back to the valuation date, then adjust spot. For the price at initiation, discount to time 0. For a later value, include only the cash flows not yet paid.
Adding income to spot and subtracting costs, which reverses the signs.
Students think of benefits as 'adding value'.
Fix: The forward holder does not receive the income, so income lowers the forward price. Storage costs raise it. Remember: benefits down, costs up.
Using the wrong time period, such as T instead of T − t in the valuation formula.
The original maturity is easy to copy from the stem.
Fix: Read the stem for 'after X months'. Remaining time is T − t. Draw a quick timeline.
Mixing discrete and continuous compounding, or forgetting the sign for the short side.
The rate is quoted in one style and the formula is memorised in another.
Fix: Match the formula to the rate stated. Then state whose position you are valuing and flip the sign for the short.

Worked examples

Example 1

A stock trades at €80. The risk-free rate is 5% per year (annual compounding). The stock pays a single dividend of €2 in six months. What is the no-arbitrage price of a one-year forward contract on the stock? A) €81.90 B) €81.95 C) €84.00

Show the solution

Identify: price at initiation, T = 1, dividend of €2 at t = 0.5.
PV of the dividend = 2 ÷ 1.05^0.5 = 2 ÷ 1.024695 = €1.9518.
Adjusted spot = 80 − 1.9518 = €78.0482.
Grow to T: F0 = 78.0482 × 1.05 = €81.9506, about €81.95.
Check the traps: option C ignores the dividend (80 × 1.05 = 84.00). Option A subtracts the dividend without discounting ((80 − 2) × 1.05 = 81.90).

Answer: B) €81.95

Example 2

A long forward on a non-dividend-paying stock was initiated at a forward price of 102.00 with one year to expiration. Three months later, the stock trades at 105.00 and the risk-free rate is 4% (annual compounding). What is the value of the long forward position now? A) 3.00 B) 5.96 C) 6.92

Show the solution

Identify: value during the life. t = 0.25, T = 1, so T − t = 0.75 years. No income or costs, so adjusted spot = 105.00.
Compute 1.04^0.75 = e^(0.75 × ln 1.04) = e^(0.029416) = 1.029852.
Discount the original forward price: 102.00 ÷ 1.029852 = 99.0434.
Vt = 105.00 − 99.0434 = 5.9566, about 5.96.
Check the traps: option A is St − F0 with no discounting (105 − 102). Option C discounts over a full year at 1.04 (102 ÷ 1.04 = 98.0769, giving 6.92), which uses the wrong time period.

Answer: B) 5.96 (a gain for the long; the short's value is −5.96)

Exam tips

Every option set has three values. One is usually the undiscounted shortcut, such as St − F0, and another uses the wrong time period. Compute the right answer, then check it against these traps.
Read whether the question asks for a price or a value. Words like 'no-arbitrage price at initiation' mean F0. 'Value of the position' or 'mark to market' mean Vt.
Look at the dates of income and costs. Only cash flows still to come after time t belong in a value calculation.
If the stem gives a continuously compounded rate, switch to e^(rT) at once. If it gives an annual rate, use (1 + r)^T.
At expiration there is no calculation beyond ST − F0 for the long. Do not waste time discounting.

Practice questions from Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Pricing and Valuation of Forward Contracts in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Pricing and Valuation of Forward Contracts: frequently asked questions

What is the difference between forward price and forward value?

The forward price is the delivery price agreed at initiation and stays fixed for that contract. The forward value is the contract's current worth, which is zero at initiation and then changes as spot and time move. Think of price as the contract term and value as the profit or loss if you closed it today.

How do I value a forward contract after initiation?

Adjust the current spot for any remaining income or carry costs, then subtract the original forward price discounted at the risk-free rate over the remaining time. That gives the value to the long. The short's value is the negative of this.

How does continuous compounding change the forward price formula?

The formula becomes F0 = S0 × e^(rT). If the asset pays a continuous yield q, use F0 = S0 × e^((r − q)T). For the value later, discount F0 with e^(−r(T − t)).

Why do dividends lower the forward price?

The holder of the forward does not receive the dividends, while the owner of the stock does. The carry strategy therefore costs less after counting the income, so the no-arbitrage forward price is lower.