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FRM Exam Part I · Pricing Financial Forwards and Futures

Forward Price Formula for No-Income Assets: Cost of Carry

Updated 11 October 2026 · Fact-checked

The forward price of an asset that pays no income is the spot price compounded at the risk-free rate to delivery: F₀ = S₀ × e^(rT) with continuous compounding, or S₀ × (1 + r)^T with annual compounding. Any other price lets you earn a risk-free profit by cash and carry or reverse cash and carry arbitrage.

Understand Cost of Carry and Forward Pricing Without Income

A forward contract fixes today the price you will pay for an asset at a future date T. The forward price is the delivery price that makes the contract worth zero at the start. It is not a forecast of the future spot price. It comes from no-arbitrage logic.

Take an asset that pays no income and costs nothing to store, such as a non-dividend-paying stock or a zero-coupon bond. Suppose you want the asset at time T. You have two ways. One: enter a forward and pay F₀ at T. Two: buy it now for S₀, financed by borrowing at the risk-free rate. Both give you the asset at T. So they must cost the same at T. Borrowing S₀ means you owe S₀ × e^(rT) at T. Therefore F₀ = S₀ × e^(rT).

The extra amount, F₀ minus S₀, is the cost of carry. With no income and no storage cost, the only carry cost is interest on the money tied up. So the forward price is always above spot when r is positive, and the gap widens with time and with the rate.

If the market price differs, arbitrage appears. If F is too high, do cash and carry: borrow S₀, buy the asset, sell the forward. At T you deliver the asset, receive F, repay S₀ × e^(rT), and keep the difference. If F is too low, do reverse cash and carry: short the asset, invest the proceeds at r, buy the forward. At T you collect S₀ × e^(rT), pay F, take delivery and close the short. Profit is S₀ × e^(rT) − F.

The price depends on the spot, the rate and the time. It does not depend on expected future spot prices or on investor risk appetite. Reverse cash and carry needs the asset to be shortable. Real markets also have costs and borrowing spreads, so the formula holds exactly only in the idealised setting GARP uses.

Key formulas to remember

Forward price, continuous compounding
F₀ = S₀ × e^(rT)
r is the continuously compounded risk-free rate, T in years. No income, no storage cost.
Forward price, discrete compounding
F₀ = S₀ × (1 + r)^T
Use when r is quoted as an annual effective rate. With m compounding periods per year: S₀ × (1 + r/m)^(mT).
Value of an existing long forward
f = S₀ − K × e^(−rT)
K is the delivery price fixed in the contract and T is the remaining life. At initiation K = F₀ so f = 0.
Cost of carry
F₀ − S₀ = S₀ × (e^(rT) − 1)
With no income, carry is pure financing cost.
Arbitrage profit at T
If F > S₀e^(rT): profit = F − S₀e^(rT). If F < S₀e^(rT): profit = S₀e^(rT) − F
Profit at delivery date, risk-free, no initial outlay.

How to solve Cost of Carry and Forward Pricing Without Income questions

Use this method for any question on no-income forward pricing, whether it asks for the fair price, an arbitrage profit or a contract value.

  1. 1Identify the asset and confirm it pays no income and has no storage cost.
  2. 2Write down S₀, the risk-free rate r and the time T in years (months ÷ 12).
  3. 3Check the compounding basis. Continuous rate: use e^(rT). Annual or periodic rate: use (1 + r/m)^(mT).
  4. 4Compute the theoretical forward price F₀ = S₀ × e^(rT).
  5. 5Compare it with the quoted forward price. If they are equal, there is no arbitrage.
  6. 6If the quote is higher, state cash and carry: borrow, buy spot, sell forward. If lower, state reverse cash and carry: short spot, invest, buy forward.
  7. 7Compute the profit at T as the absolute difference between the quote and the theoretical price. Discount it by e^(−rT) if the question asks for present value.
  8. 8For the value of an existing forward, use S₀ − K × e^(−rT) with the remaining time.

Quickest way: Compound the spot, then compare

When to use it: Use when you have a single numeric forward pricing or arbitrage question and little time.

  1. Convert T to years and read r as continuous unless told otherwise.
  2. Calculate e^(rT) on your calculator (press e^x key). For small rT, e^(rT) ≈ 1 + rT as a sanity check.
  3. Multiply by S₀ to get F₀.
  4. If an option list is given, eliminate answers below S₀ first, because F₀ must exceed S₀ when r > 0.
  5. For arbitrage direction, remember: quote too high means sell forward and buy spot; quote too low means buy forward and short spot.

Common mistakes in Cost of Carry and Forward Pricing Without Income

  • Using months directly as T, for example T = 6 instead of 0.5.

    The question gives maturity in months and the rate is annual, so students rush.

    Fix: Always convert to years first. Write T = months ÷ 12 before touching the formula.

  • Applying e^(rT) when the rate is quoted as an annual effective rate, or the reverse.

    Students ignore the compounding basis stated in the question.

    Fix: Read the compounding wording. Use (1 + r)^T for annual rates and e^(rT) for continuous rates. If they differ, the answers differ slightly and options are built to catch this.

  • Treating the forward price as the expected future spot price.

    The word forward sounds like a forecast.

    Fix: Remember the forward price comes only from S₀, r and T by no-arbitrage. Expectations do not enter.

  • Getting the arbitrage direction backwards.

    Students memorise the names but not the logic.

    Fix: Ask which side is overpriced. If the forward is too expensive, sell it and buy the asset cheaply now, funding with a loan.

  • Using the forward price instead of the delivery price K when valuing an existing contract.

    Both are called forward prices and look alike.

    Fix: Value = S₀ − K × e^(−rT), where K is the contract's fixed price and T is the time left. Do not use F₀ as K.

  • Adding dividends or storage costs to a no-income asset, or ignoring them when stated.

    Students apply one formula to every asset.

    Fix: Check the wording first. This formula applies only when there is no income and no storage cost.

Worked examples

Example 1

A non-dividend-paying stock trades at $80. The continuously compounded risk-free rate is 5% per year. What is the fair 9-month forward price?

Show the solution
  1. Identify: no income, so F₀ = S₀ × e^(rT).
  2. S₀ = 80, r = 0.05, T = 9 ÷ 12 = 0.75.
  3. rT = 0.05 × 0.75 = 0.0375.
  4. e^0.0375 ≈ 1.03821.
  5. F₀ = 80 × 1.03821 = 83.06.

Answer: The fair forward price is about $83.06.

Example 2

A stock trades at $50 and pays no dividends. The continuously compounded risk-free rate is 4% per year. A 1-year forward is quoted at $53. Is there an arbitrage, and what is the profit at delivery?

Show the solution
  1. Theoretical price: F₀ = 50 × e^0.04.
  2. e^0.04 ≈ 1.04081, so F₀ = 52.04.
  3. Quoted forward $53 is higher than $52.04, so the forward is overpriced.
  4. Strategy (cash and carry): borrow $50 at 4%, buy the stock, sell the forward at $53.
  5. At T: deliver the stock and receive $53. Repay the loan of 50 × e^0.04 = $52.04.
  6. Profit = 53 − 52.04 = $0.96.

Answer: Yes. Do cash and carry for a risk-free profit of about $0.96 per share at delivery.

Exam tips

  • Read the compounding basis in the question before calculating. It is the most common trap in the options.
  • Use the sanity check that F₀ > S₀ when r > 0 and there is no income to rule out wrong options quickly.
  • Arbitrage questions usually ask for the strategy or the profit. Practise stating the three legs: borrow or invest, spot trade, forward trade.
  • Know that the forward price at initiation gives zero value, and that the value later depends on S₀ and K, not on F₀.
  • Practise e^x on your approved calculator so each calculation takes seconds.

Practice questions from Pricing Financial Forwards and Futures

Cost of Carry and Forward Pricing Without Income in other exams

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

Cost of Carry and Forward Pricing Without Income: frequently asked questions

What is the forward price formula for an asset with no income?

It is F₀ = S₀ × e^(rT) with continuous compounding, or S₀ × (1 + r)^T with annual compounding. S₀ is the spot price, r the risk-free rate and T the time to delivery in years.

What is cash and carry arbitrage?

It applies when the forward price is above S₀ × e^(rT). You borrow money, buy the asset and sell the forward. At delivery you hand over the asset, receive the forward price, repay the loan and keep the risk-free profit.

Does the forward price depend on the expected future spot price?

No. It depends only on the spot price, the risk-free rate and time, because arbitrage forces that relationship. Expected future spot prices and risk preferences do not enter the formula.

How is the value of an existing forward different from its forward price?

The forward price is the delivery price that gives zero value when the contract starts. Later, the long forward is worth S₀ − K × e^(−rT), where K is the original delivery price and T is the remaining time.