FRM Exam Part I · Pricing Financial Forwards and Futures

Valuing Forwards and Futures Prices vs Expected Spot

Updated 11 October 2026 · Fact-checked

The value of an existing long forward is f = S₀ − K·e^(−rT) for an asset with no income, which equals the present value of (F₀ − K). Forward and futures prices are equal when rates are constant. Futures prices are not unbiased predictors of future spot prices in general.

Understand Futures Prices, Expected Spot Prices and Valuation of Forwards

A forward contract fixes a delivery price K today for an asset bought at time T. When you enter it at the fair forward price, its value is zero. As time passes, the spot price and interest rates move. The forward price for new contracts changes, but your K stays fixed. So your contract now has a positive or negative value.

The logic is simple. A long forward at K gives you the asset at T and costs you K at T. The value today is the present value of the asset you receive minus the present value of K. If the asset pays no income, the present value of the asset is the spot price S₀. So f = S₀ − K·e^(−rT).

There is a second way to see it. The current forward price F₀ is the delivery price that gives a zero value today. A contract with delivery price K differs from it by (F₀ − K) at time T. Discount that difference: f = (F₀ − K)·e^(−rT). This version works for any asset, with or without income, as long as you have F₀.

Forward and futures prices are the same if interest rates are constant and the same for all maturities. Futures are settled daily, so gains and losses are reinvested or financed at changing rates. When rates are stochastic, the sign of the correlation between the futures price and interest rates matters. If futures prices are positively correlated with rates, a long futures holder gains when rates are high and can invest those gains at high rates. Losses come when rates are low and are financed cheaply. So futures prices tend to be slightly higher than forward prices. With negative correlation the opposite holds. For short maturities the difference is small.

The forward price is not a forecast of the future spot price. The cost-of-carry argument is based on arbitrage, not on expectations. Whether the forward or futures price is above or below the expected spot price depends on the systematic risk of the underlying. For an asset with no income or storage costs, F₀ = E(S_T)·e^((r − k)T), where k is the return investors require on the asset, set by its systematic risk (as in CAPM). If the systematic risk is positive, k > r and F₀ < E(S_T). If the systematic risk is zero, k = r and F₀ = E(S_T). If the systematic risk is negative, k < r and F₀ > E(S_T).

In the hedging-pressure theory, normal backwardation means the futures price is below the expected future spot price. It arises when hedgers are mostly short futures, so speculators must be long and need a positive expected return. Normal contango is the opposite case: the futures price is above the expected future spot price, which arises when hedgers are mostly long futures.

Do not confuse these with contango and backwardation in the usual curve sense. There, contango means the futures price is above the current spot price, so futures prices rise with maturity. Backwardation means the futures price is below the current spot price. The curve-shape terms compare futures with today's spot. The normal versions compare futures with the expected future spot. Keep the two ideas separate.

Key formulas to remember

Value of long forward, no income: f = S₀ − K·e^(−rT); Continuous compounding. For a short forward, the value is the negative of this.
Value of long forward using the forward price: f = (F₀ − K)·e^(−rT); Works for any underlying. F₀ is today's forward price for the same maturity.
Forward price, no income: F₀ = S₀·e^(rT); Cost-of-carry result, so f = 0 when K = F₀.
Value with known income (PV of income I): f = S₀ − I − K·e^(−rT); I is the present value of income during the life of the contract.
Value with continuous yield q: f = S₀·e^(−qT) − K·e^(−rT); Use for stock indices and foreign currencies, where q is the dividend yield or the foreign rate.
Futures vs expected spot (systematic risk): F₀ = E(S_T)·e^((r − k)T). If k > r: F₀ < E(S_T). If k = r: F₀ = E(S_T). If k < r: F₀ > E(S_T).; k is the required return on the asset, set by its systematic risk as in CAPM. Applies to an asset with no income or storage costs. Positive systematic risk means k > r.

How to solve Futures Prices, Expected Spot Prices and Valuation of Forwards questions

Use this order for any question on valuing a forward or comparing forward, futures and expected spot prices.

1Identify the position (long or short) and the asset type: no income, known cash income, or continuous yield.
2Write down S₀, K, r, T, and any income I or yield q. Convert T to years and check that r uses the compounding the question gives.
3Choose the method: use f = S₀ − I − K·e^(−rT) if you have the spot price, or f = (F₀ − K)·e^(−rT) if the question gives the current forward price.
4Compute the present value of the income or the yield-adjusted spot price first, then discount K.
5Subtract to get the long value. Flip the sign if the position is short.
6For forward vs futures questions, check whether rates are constant (prices equal) or stochastic (look at the correlation sign).
7For futures vs expected spot questions, find the sign of the asset's systematic risk and apply the ordering above.

Quickest way: Discount the price gap

When to use it: Use it when the question gives both the old delivery price K and a current forward price or enough data to find it quickly.

Find the current forward price F₀ with S₀·e^((r − q)T) or S₀·e^(rT).
Take F₀ − K. Positive means a gain for the long.
Multiply by e^(−rT) using your calculator's e^x key.
Check that the sign makes sense: if spot rose since entry, the long should have a gain.

Common mistakes in Futures Prices, Expected Spot Prices and Valuation of Forwards

Valuing the forward as S₀ − K with no discounting.
It looks like a payoff at maturity, and students forget K is paid at time T.
Fix: Always discount K, or discount F₀ − K, back from T to today.
Saying a new forward has value equal to the forward price.
Price and value are confused.
Fix: The forward price is the delivery price K that makes the contract's value zero at inception. A new forward has value zero; its price is a delivery price, not a value.
Forgetting to subtract the PV of income or to use S₀·e^(−qT).
Students memorize the no-income formula.
Fix: Read the question for dividends, coupons or a foreign rate before you pick the formula.
Treating the forward price as the best forecast of the spot price.
Forward prices look like market expectations.
Fix: Forward prices come from arbitrage. They equal expected spot only when the asset has no systematic risk.
Getting the sign of the futures-forward difference wrong under stochastic rates.
Students memorize the answer without the logic.
Fix: Ask who gains when rates are high. Positive correlation between futures price and rates means futures price is above the forward price.
Not flipping the sign for a short position.
Students rush and report the long value.
Fix: Short value = −(long value). Write long or short next to your answer.

Worked examples

Example 1

A stock pays no dividends. You entered a long forward 6 months ago with delivery price K = USD 50 and original maturity of 1 year. Now the stock price is USD 54, the remaining time is 0.5 years and the risk-free rate is 4% continuously compounded. What is the value of the long forward?

Show the solution

The asset has no income, so f = S₀ − K·e^(−rT).
S₀ = 54, K = 50, r = 0.04, T = 0.5 (the remaining time).
e^(−0.04 × 0.5) = e^(−0.02) = 0.980199.
K·e^(−rT) = 50 × 0.980199 = 49.0099.
f = 54 − 49.0099 = 4.9901.

Answer: The long forward is worth about USD 4.99.

Example 2

An index has a continuous dividend yield of 2% and the risk-free rate is 5%, both continuously compounded. The index level is 3,000. You are long a 9-month forward on the index with delivery price 3,000. What is the value of your forward, and what is the current forward price?

Show the solution

T = 0.75, r = 0.05, q = 0.02, S₀ = 3,000, K = 3,000.
Forward price F₀ = S₀·e^((r − q)T) = 3,000 × e^(0.03 × 0.75) = 3,000 × e^(0.0225).
e^(0.0225) = 1.022755, so F₀ = 3,068.27.
Value f = (F₀ − K)·e^(−rT) = 68.27 × e^(−0.0375).
e^(−0.0375) = 0.963194, so f = 68.27 × 0.963194 = 65.76.
Check with the other formula: 3,000·e^(−0.015) − 3,000·e^(−0.0375) = 3,000 × (0.985112 − 0.963194) = 3,000 × 0.021918 = 65.75, which agrees up to rounding.

Answer: The forward price is about 3,068.27 and the long forward is worth about 65.76 index points.

Exam tips

Expect a numeric question where you must pick between S₀ − K·e^(−rT) and (F₀ − K)·e^(−rT). Know that they are the same thing.
Read for income, dividends or a foreign rate. These change the formula and are the usual trap.
Conceptual questions test the sign: positive correlation between rates and futures prices means futures price above forward price.
For futures vs expected spot, link the answer to systematic risk (beta). Zero beta gives equality.
Do not round e^x values early. Keep at least five decimals to match the answer options.

Practice questions from Pricing Financial Forwards and Futures

Futures Prices, Expected Spot Prices and Valuation of Forwards in other exams

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

Futures Prices, Expected Spot Prices and Valuation of Forwards: frequently asked questions

What is the formula for the value of a forward contract?

For a long forward on an asset with no income, f = S₀ − K·e^(−rT). More generally, f = (F₀ − K)·e^(−rT), where F₀ is the current forward price. A short forward has the opposite sign.

Are futures prices unbiased predictors of the future spot price?

Not in general. The futures price equals the expected spot price only if the asset has zero systematic risk. With positive systematic risk the futures price is below the expected spot price, and with negative systematic risk it is above.

When are forward and futures prices equal?

They are equal in theory when interest rates are constant and the same for all maturities. With stochastic rates they can differ because futures are settled daily. The gap is usually small for short maturities.

How do I value an existing forward contract if the asset pays dividends?

Subtract the present value of the income from S₀, then subtract the present value of K. With a continuous yield, use f = S₀·e^(−qT) − K·e^(−rT). You can also find F₀ first and use (F₀ − K)·e^(−rT).