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CFA Level I Exam · Pricing and Valuation of Futures Contracts

Pricing of Forward and Futures Contracts: Cost of Carry

Updated 7 October 2026 · Fact-checked

The cost-of-carry model sets the forward price so that buying the asset now and holding it gives the same result as agreeing to buy later. With no income: F0 = S0 × (1 + r)^T. Subtract the future value of income; add the future value of storage costs. Convenience yield is a rate that lowers the price.

Understand Pricing of Forward and Futures Contracts

A forward contract is an agreement to buy an asset at a fixed price on a future date. Its price is set by no-arbitrage. You could buy the asset today with borrowed money and hold it until delivery. Or you could enter the forward and wait. Both must cost the same, or someone earns a risk-free profit.

For an asset with no income or costs, holding it costs only the interest you give up. So the forward price is the spot price grown at the risk-free rate: F0 = S0 × (1 + r)^T. The difference between F0 and S0 is the cost of carry.

Income from the asset lowers the cost of carry. If you hold a stock paying dividends, you receive cash that the forward buyer does not. So you subtract the future value of those benefits at the forward's expiry. Storage costs raise the cost of carry, so you add their future value. Think of it as: forward price = future value of (spot + costs - benefits).

Convenience yield is a non-monetary benefit of physically holding a commodity, such as keeping a factory running during a shortage. It lowers the forward price, but it is not a cash flow with a payment date, so you do not compound it forward like a dividend or coupon. It is usually modelled as a continuous rate (the δ in e^((r − δ + c)T)) or as an implied adjustment. It is not observed directly. It is the adjustment that makes the actual forward price lower than the pure carry price.

Forwards and futures usually have almost the same price. Futures are marked to market daily and margined. Differences arise mainly when interest rates are correlated with the futures price. If they are uncorrelated, the two prices are equal. If futures prices are positively correlated with rates, the futures price tends to be higher than the forward price. If negatively correlated, it tends to be lower. Interest-rate effects on daily margin flows drive this.

Key formulas to remember

Forward price, no income or costs (discrete)
F0(T) = S0 × (1 + r)^T
r is the annual risk-free rate; T is in years. Exam questions often state which compounding to use.
Forward price, continuous compounding
F0(T) = S0 × e^(rT)
Use the e^x key when the rate is stated as continuously compounded.
Forward price with income and costs
F0(T) = (S0 − PV of benefits + PV of costs) × (1 + r)^T, or F0(T) = S0 × (1 + r)^T − FV of benefits + FV of costs
Benefits here are dated cash flows such as dividends and coupons. Costs are storage and insurance. Both forms give the same answer. Convenience yield is not a dated cash flow, so it does not go in these terms; model it as a continuous rate (next formula).
Forward price with continuous yield
F0(T) = S0 × e^((r − δ + c)T)
δ is the continuous income yield or convenience yield; c is the continuous storage cost rate.
Arbitrage rule
If market F0 > model F0: sell forward, buy asset with borrowed funds. If market F0 < model F0: buy forward, short the asset and invest proceeds.
Either mispricing locks in a risk-free profit equal to the difference at expiry.
Futures vs forward
Uncorrelated rates: futures price = forward price
Positive correlation of futures price and rates favours a higher futures price; negative correlation favours a lower one.

How to solve Pricing of Forward and Futures Contracts questions

Use this order for any forward or futures pricing question.

  1. 1Identify the underlying and list what it pays or costs to hold: dividends, coupons, storage, insurance, convenience yield.
  2. 2Note the time to expiry T in years and the risk-free rate r. Check the compounding stated.
  3. 3Decide the form: discrete (1 + r)^T or continuous e^(rT).
  4. 4Compute the future value of benefits and costs at expiry, or their present value today, using the same r. Do not mix the two approaches.
  5. 5Apply F0 = S0 × (1 + r)^T − FV(benefits) + FV(costs). Or use the present-value form on the net spot.
  6. 6If the question gives a market price, compare it with your model price. Above the model: sell forward and buy spot. Below: buy forward and short spot.
  7. 7Check that your answer is sensible. Income should lower F0 relative to the no-income case, and costs should raise it.

Quickest way: Net-carry shortcut

When to use it: Use when the question asks for a forward price with a small number of income or cost items and the options are close together.

  1. Compute the no-income price S0 × (1 + r)^T first.
  2. Estimate each benefit's future value as benefit × (1 + r)^(time remaining after payment).
  3. Subtract benefits and add costs.
  4. Compare with the three options. The wrong options often ignore compounding on income or use the present value without growing it.

Common mistakes in Pricing of Forward and Futures Contracts

  • Subtracting the present value of dividends from the future price instead of the future value.

    Mixing the two equivalent forms of the formula.

    Fix: Subtract PV from spot before compounding, or subtract FV after compounding. Never subtract PV from an already compounded price.

  • Growing a dividend for the full term T.

    Treating every cash flow as if it arrived today.

    Fix: A dividend received at time t is compounded only for T − t, the time left to expiry.

  • Adding convenience yield to the price.

    Seeing it as a cost of holding.

    Fix: Convenience yield is a benefit. It reduces the forward price like income.

  • Using annual r for a period given in months without converting.

    Rushing and leaving T as 6 instead of 0.5.

    Fix: Always express T in years before computing.

  • Picking the wrong arbitrage direction.

    Memorizing the action rather than the logic.

    Fix: If the forward is overpriced, sell it and buy the asset. If underpriced, buy it and short the asset.

Worked examples

Example 1

A stock trades at $50. It will pay a dividend of $1 in 3 months. The risk-free rate is 4% per year, compounded annually. What is the no-arbitrage price of a 6-month forward? Options: A) $49.00, B) $49.98, C) $50.99

Show the solution
  1. T = 0.5. Spot grown to expiry: 50 × 1.04^0.5 = 50 × 1.019804 = 50.9902.
  2. The dividend is received at month 3, so it grows for 0.25 years: 1 × 1.04^0.25 = 1.009853.
  3. F0 = 50.9902 − 1.0099 = 49.9803.
  4. Check with the PV form: PV of dividend = 1 ÷ 1.04^0.25 = 0.99028. Net spot = 49.00972. Grow: 49.00972 × 1.019804 = 49.9802.
  5. Both methods give about $49.98, which is option B.

Answer: B) $49.98. The other options come from common errors: A ($49.00) ignores interest altogether (50 − 1), and C ($50.99) ignores the dividend. Subtracting the dividend without compounding it gives about $49.99, only one cent away from the correct value, so a well-built exam item would not offer both.

Example 2

A commodity trades at $80. Storage costs are $2 per year paid at the end of the year. Convenience yield is ignored. The risk-free rate is 5% per year. What is the 1-year forward price? Options: A) $82.00, B) $84.00, C) $86.00

Show the solution
  1. Spot grown for one year: 80 × 1.05 = 84.00.
  2. Storage cost is paid at the end of the year, so its future value at expiry is $2.00.
  3. F0 = 84.00 + 2.00 = 86.00.
  4. Option A forgets interest; option B forgets storage.

Answer: $86.00, option C.

Exam tips

  • Check the timing of every cash flow. Questions often give dividends at different dates to test compounding.
  • When options are listed small to large, eliminate those that move in the wrong direction: income should lower the price and storage should raise it.
  • Know the arbitrage direction cold. Many items ask what a trader should do if the forward trades above or below the model price.
  • Continuous-compounding questions use e^x. On the BA II Plus, enter r × T (for example 0.05 × 0.5 = 0.025), then press 2ND, then LN to get e^x.
  • Remember futures versus forward: the prices are equal when interest rates are uncorrelated with futures prices.

Practice questions from Pricing and Valuation of Futures Contracts

Pricing of Forward and Futures 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 of Forward and Futures Contracts: frequently asked questions

What is the cost-of-carry model?

It prices a forward by adding the net cost of holding the asset until expiry to the spot price. Interest and storage add to cost, while income and convenience yield reduce it. The result is the price that leaves no arbitrage.

How do I calculate a forward price with dividends?

Grow the spot price at the risk-free rate to expiry and subtract the future value of the dividends. Or subtract the present value of the dividends from spot first and then grow the result. Both give the same answer.

What is convenience yield and how does it affect the price?

It is the non-monetary benefit of holding the physical commodity, such as protection against shortages. It works like income, so it lowers the forward price. It cannot be observed directly.

Is a forward price the same as a futures price?

Not always, but they are close. They are equal when interest rates are uncorrelated with futures prices. Daily marking to market in futures creates differences when rates and futures prices move together.