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

Cost of Carry Model for Forward and Futures Prices

Updated 7 October 2026 · Fact-checked

The cost of carry is the net cost of holding the underlying until the contract expires. Forward price = (spot + PV of carry costs − PV of carry benefits) × (1 + r)^T. Costs such as storage and interest raise the price. Benefits such as dividends and convenience yield lower it.

Understand Cost of Carry: Benefits and Costs of Holding the Underlying

A forward contract lets you buy an asset later at a price fixed today. You could get the same asset by buying it now and holding it. The forward price must therefore match the cost of that alternative. If it does not, an arbitrage profit exists.

Holding the asset has costs and benefits. Costs include the interest you give up on the money tied up in the asset (the financing cost), and for physical goods, storage and insurance. Benefits include dividends on equities, coupons on bonds, and the convenience yield on commodities. Net of these, you get the cost of carry.

The forward buyer does not hold the asset, so does not get the benefits or pay the costs. The forward buyer does not receive the benefits, so the price is reduced by them. The buyer also avoids the costs of holding the asset, so the price is increased by them. Higher interest rates and storage costs push the forward price above spot. Higher dividends and convenience yield pull it down.

Convenience yield is the non-monetary benefit of physically holding a commodity, such as keeping a refinery running during a shortage. It cannot be observed directly. It is usually inferred from market prices. The forward price can be below spot if the convenience yield is large. The CFA curriculum treats it as a benefit of holding the underlying, which lowers the forward price.

Two versions of the math appear. In discrete form you subtract the future value or present value of cash flows. In continuous form, you use exponentials with a yield. Both rest on the same no-arbitrage idea. A third shortcut applies when carry is a known percentage of spot.

Key formulas to remember

Forward price, general (PV form)
F₀ = (S₀ + PVC − PVB) × (1 + r)^T
PVC = present value of carry costs (e.g. storage); PVB = present value of carry benefits (e.g. dividends, convenience yield). Use the same r and T throughout.
Forward price, future value form
F₀ = S₀ × (1 + r)^T + FVC − FVB
FVC and FVB are values at the contract expiry date. Use this when costs or benefits are already given at expiry.
Forward price, no carry
F₀ = S₀ × (1 + r)^T
Applies when the underlying pays no income and has no holding costs.
Continuous compounding with yield
F₀ = S₀ × e^((r − γ) × T)
This is the standard curriculum form. r is the continuously compounded risk-free rate and γ is the continuously compounded benefit yield (dividend yield or convenience yield). If a storage cost rate c is also given as a continuous rate, treat it as a negative benefit, so the net yield is γ − c and the exponent becomes (r − γ + c) × T. Use when the question states continuous rates.
Value of carry sign rule
Costs ↑ → F₀ ↑; Benefits ↑ → F₀ ↓
Use to check direction and to eliminate wrong options.

How to solve Cost of Carry: Benefits and Costs of Holding the Underlying questions

Use this routine for any question on forward pricing with holding costs or benefits.

  1. 1Identify the underlying, spot price S₀, risk-free rate r and time T in years. Convert months to years (for example 6 months = 0.5).
  2. 2List every carry cost (storage, insurance) and every carry benefit (dividends, coupons, convenience yield). Note the timing of each.
  3. 3Check the compounding. Discrete annual rates use (1 + r)^T. Continuous rates use e^(rT).
  4. 4Bring each cash flow to the same date. Discount to today for the PV form, or compound to expiry for the FV form.
  5. 5Apply the formula: add carry costs and subtract carry benefits, then compound the net spot cost to expiry.
  6. 6Check direction. Benefits should lower F₀ relative to the no-carry price. Costs should raise it.
  7. 7If the question gives a market forward price, compare it with your fair price. If market is higher, sell forward and buy spot. If lower, buy forward and short spot.

Quickest way: Compare to the no-carry price, then adjust

When to use it: Use on multiple-choice items when the options are far apart and you must save time.

  1. Compute the no-carry price S₀ × (1 + r)^T first. This is your anchor.
  2. Decide the direction. If the question has only benefits, the answer is below the anchor. If only costs, above.
  3. Compute the compounded value of each carry item and apply it to the anchor.
  4. Match to the options. Options are listed smallest to largest, so the direction check often removes two choices at once.

Common mistakes in Cost of Carry: Benefits and Costs of Holding the Underlying

  • Adding dividends instead of subtracting them

    Students think of dividends as a plus for the holder and add them.

    Fix: The forward buyer misses the dividends, so the forward price is lower. Always subtract benefits and add costs.

  • Mixing PV and FV amounts

    A dividend is given at expiry but is subtracted inside the bracket that is later compounded.

    Fix: Put every item on one date. Discount dividends to today before subtracting inside the bracket, or compound them to expiry (FV) before subtracting after compounding the spot.

  • Forgetting to compound storage costs paid in advance

    Students add the storage cost to spot without checking when it is paid.

    Fix: If storage is paid at time 0, add it to spot before compounding. If paid later, discount it to today or compound the others to match.

  • Using months as years

    Rushing under time pressure, T = 6 is typed instead of 0.5.

    Fix: Write T in years on the first line of your working.

  • Assuming convenience yield always applies to financial assets

    It is memorised as a general carry benefit.

    Fix: Convenience yield belongs to physical commodities. Equities have dividends, bonds have coupons.

  • Believing the forward price must exceed spot

    Positive interest makes it seem automatic.

    Fix: If benefits exceed financing plus storage costs, the forward price is below spot.

Worked examples

Example 1

A stock trades at €80. It will pay a dividend of €2 in 3 months. The annual risk-free rate is 4% with annual compounding. What is the fair price of a 6-month forward contract on the stock? Options: A) €79.56 B) €80.99 C) €82.37

Show the solution
  1. S₀ = 80, T = 0.5, r = 4%. Dividend €2 at t = 0.25.
  2. PV of dividend = 2 ÷ 1.04^0.25. 1.04^0.25 = 1.009853. PV = 2 ÷ 1.009853 = 1.9805.
  3. Net spot = 80 − 1.9805 = 78.0195.
  4. Compound to 6 months: 1.04^0.5 = 1.019804.
  5. F₀ = 78.0195 × 1.019804 = 79.5646, about €79.56.
  6. Direction check: the no-carry price is 80 × 1.019804 = 81.58. The dividend lowers the price below this no-carry value, so the answer is the option equal to €79.56.

Answer: €79.56, which is option A.

Example 2

Gold spot is $2,000 per ounce. Storage and insurance cost $12 per ounce per year, paid at the end of the year. The annual risk-free rate is 5% and no other carry items apply. What is the 1-year forward price? Options: A) $2,000 B) $2,100 C) $2,112

Show the solution
  1. Spot compounded: 2,000 × 1.05 = 2,100.
  2. Storage is paid at the end of the year, which is the forward expiry. FV of cost = $12 with no extra compounding.
  3. F₀ = 2,100 + 12 = $2,112.
  4. Direction check: costs raise the price above 2,100, so $2,100 and $2,000 are eliminated.

Answer: $2,112 per ounce, which is option C.

Exam tips

  • Draw a one-line timeline with spot, cash flows and expiry before you calculate. Most errors come from timing.
  • Use the direction rule to eliminate two options. Costs push up, benefits push down.
  • Read whether rates are discrete or continuous. The same numbers give different answers.
  • Expect conceptual items on convenience yield: it is a benefit to the holder of the physical asset, it is not directly observable, and it lowers the forward price.
  • On a TI BA II Plus, compute 1.04^0.5 with 1.04 then yˣ then 0.5 then =. On an HP 12C, enter 1.04, ENTER, 0.5, yˣ.

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

Cost of Carry: Benefits and Costs of Holding the Underlying: frequently asked questions

What is the cost of carry in simple terms?

It is the net cost of holding the underlying until the forward expires. It includes financing and storage costs, minus benefits such as dividends and convenience yield. A positive net cost raises the forward price above spot.

What is convenience yield?

Convenience yield is the non-monetary benefit of holding a physical commodity instead of a forward contract on it. It matters when supply is tight or production must continue. It lowers the forward price and cannot be observed directly.

How do you calculate a forward price with dividends and storage costs?

Add the present value of storage costs to spot and subtract the present value of dividends. Then compound the result at the risk-free rate over the contract term. If the amounts are given at expiry, adjust them after compounding instead.

Can a forward price be lower than the spot price?

Yes. If carry benefits such as dividends or convenience yield are larger than the financing and storage costs, the fair forward price falls below spot. This is common in commodity markets under shortage.