Lesson 3 of 4 6–9% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Cost of carry: income, storage and convenience yield

Holding an asset costs the interest forgone plus any storage or insurance, and may pay income or a convenience benefit. The forward price adds the costs and subtracts the benefits: F₀(T) = [S₀ − PV(I) + PV(C)](1 + r)^T, where PV is present value. It is a no-arbitrage price, not a forecast.

8 min cram, 36 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 4. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Replication and the no-arbitrage forward price, Time Value of Money in Finance

On the exam

Explain the difference between the spot and expected future price of an underlying and the cost of carry associated with holding the underlying asset

Explain the difference between the spot and the expected future price, and the cost of carry. Expect a forward price with a dividend, storage cost or yield, whether F₀(T) is above or below S₀, and what a convenience yield is.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

Why is the spot price not the expected future price?

The spot price is what the asset costs today. The expected future price , E(S_T), is what investors think it will be worth at T. The two differ because investors who hold a risky asset demand a reward for the risk. The spot price is the expected future price discounted at a rate that includes a risk premium , λ, on top of the risk-free rate.

Spot price from the expected future price Know the idea

S_0 = \frac{E(S_T)}{(1 + r + \lambda)^T}

A forward price is built differently. It compounds the spot price at the risk-free rate alone, because a forward can be replicated without taking any view on risk.

Exam tip Spot discounts the forecast at a risk-adjusted rate. The forward compounds spot at the risk-free rate. So F₀(T) is never a forecast of S_T.

What is the cost of carry?

The cost of carry is the net of the costs and benefits of owning the asset from today until T. It is the reason a forward price differs from spot.

What goes into the cost of carry

Cost of carrycosts and benefits of holding the asset until T

Costs Memorize push the forward price up
- Opportunity cost, r Memorize interest forgone; applies to every asset
- Storage, insurance, transport, spoilage Memorize physical commodities
Benefits Memorize pull the forward price down
- Cash income Memorize dividends, coupons, index dividend yield
- Convenience yield Memorize non-cash benefit of holding the physical commodity

The first cost is always there: the opportunity cost of the money tied up in the asset, measured by the risk-free rate r. It applies whether you borrow to buy or use your own cash. Physical commodities add storage costs , insurance, transport and spoilage. Stocks and bonds add benefits: dividend income and coupons. An equity index pays an average dividend yield , quoted as a rate.

The owner of the asset gets the benefits and pays the costs. The buyer of a forward gets neither, so the price they agree to pay reflects both. Costs raise the forward price, because a seller expects to be compensated. Benefits lower it, because the forward buyer forgoes them.

Cost of carry by asset class

Asset classwhat offsets the risk-free rate

Equities no storage cost
- Non-dividend stock none; only r
- Dividend-paying stock dividends lower F₀(T)
- Equity index dividend yield, a rate
Commodities physical holding
- Storage and insurance Memorize raise F₀(T)
- Convenience yield Memorize lowers F₀(T)
Bonds coupon income lowers F₀(T)
Foreign exchange rate differential r_f − r_d (next lesson)

How do income and costs enter the forward price?

With known amounts of income (I) and costs (C), take their present values at the risk-free rate, change S₀ by them, and compound the result.

Forward price with known income and costs Memorize

F_0(T) = [S_0 - PV_0(I) + PV_0(C)]\,(1 + r)^T

Income is subtracted and costs are added. Each amount is discounted from the date it is paid back to t = 0. Then the bracket is compounded to T.

Worked example A share with a dividend

A share trades at EUR 64. It pays a dividend of EUR 1.20 in two months. The risk-free rate is 4% and the forward expires in six months.

Present value of the dividend, two months from now: 1.20 ÷ 1.04^(2/12) = EUR 1.1922.
Subtract it from the spot price: 64 − 1.1922 = 62.8078.
Compound for six months: 62.8078 × 1.04^0.5 = EUR 64.05.

Without the dividend the forward would be 64 × 1.04^0.5 = EUR 65.27. The dividend lowers the forward price by about EUR 1.22.

Steps 1 and 3 on your calculator

1.04y^x(2÷12)=1/x×1.2= 1.1922
1.04y^x0.5=×(64−1.1922)= 64.05

y^x power · 1/x reciprocal

1.04ENTER2ENTER12÷y^x1/x1.2× 1.1922
64ENTER1.1922−1.04ENTER0.5y^x× 64.05

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Rearrange itForward price with income and storage costs: solve for any input

The question asks for

An asset trades at 64.00. The present value of the income it pays before expiration is 1.19 and the present value of its storage costs is 0.00. The risk-free rate is 4.00% and the contract runs 0.5 years. The no-arbitrage forward price is closest to:

Spot price, S₀Present value of income, PV₀(I)Present value of costs, PV₀(C)Risk-free rate, r (annual)%Time to expiration, T (years)

F_{0} (T) = [S_{0} - P V_{0} (I) + P V_{0} (C)] (1 + r)^{T}

(64.00 − 1.19 + 0.00) × (1 + 0.04)^0.5 = 64.05

Forward price, F₀(T)64.05

On your calculator

1.04y^x0.5=1.0198
×(64−1.1922+0)=64.0516

y^x power

64ENTER1.1922−0+62.8078
1.04ENTER0.5y^x×64.0516

ENTER enter (on the HP, puts the number on the stack) · y^x power

How do you price with yields and costs as rates?

For indexes, commodity indexes and currencies, benefits and costs come as a yearly rate, and the curriculum uses continuous compounding . The income yield is i and the storage cost rate is c.

Forward price with yields and costs as rates Memorize

F_0(T) = S_0\,e^{(r + c - i)T}

The three rates net out: r plus c minus i is the net cost of carry, as a rate. Positive means the forward is above spot.

Rearrange itForward price with continuous yields and costs: solve for any input

The question asks for

An index stands at 5,200.00. The continuously compounded risk-free rate is 4.50%, the storage cost rate is 0.00% and the dividend yield is 1.80%. The no-arbitrage forward price for 0.5 years ahead is closest to:

Spot price or index level, S₀Risk-free rate, r (continuous)%Storage cost rate, c (continuous)%Income or dividend yield, i (continuous)%Time to expiration, T (years)

F_{0} (T) = S_{0} e^{(r + c - i) T}

5,200.00 × e^((0.045 + 0 − 0.018) × 0.5) = 5,270.68

Forward price, F₀(T)5,270.68

On your calculator

0.045+0−0.018=×0.5=0.0135
2NDe^x×5200=5270.676

2ND second function (the shifted label above a key) · e^x e to the power x

0.045ENTER0+0.018−0.5×0.0135
ge^x5200×5270.676

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

InteractiveFrom spot to forward: the carry bridge

Try this 0 of 3

Load the tight-inventory commodity: the forward falls below spot
On the dividend share, set the dividend to zero
Spot vs expected: set the risk premium to zero

Spot S₀ (EUR)

Risk-free rate r4%

Time to expiry T6 months

Income paid before T (e.g. a dividend)

Paid after2 months

PV of storage and insurance costs

Spot S₀EUR 64.00

Forward F₀(T)EUR 64.05

F₀(T) − S₀+0.05

Interest and costs outweigh the benefits: the forward is above spot.

A convenience yield is quoted as a rate: switch to Rates (continuous) to add one.

Show working

F₀(T) = [S₀ − PV(I) + PV(C)] × (1 + r)^T

PV(I) = 1.20 ÷ 1.04^2/12 = 1.1922

F₀(T) = (64.00 − 1.1922) × 1.04^0.5 = 62.8078 × 1.019804 = EUR 64.05

Traps: subtracting the 1.20 itself gives 64.04; ignoring the income gives 65.27.

The forward on your calculator

1.04y^x(2÷12)=1/x×1.2=1.1922, PV of the income
1.04y^x0.5=×(64−1.1922)=64.05, the forward

y^x power · 1/x reciprocal

1.04ENTER2ENTER12÷y^x1/x1.2×1.1922, PV of the income
64ENTER1.1922−1.04ENTER0.5y^x×64.05, the forward

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Costs push the forward up, benefits pull it down; the risk premium never enters.

What is a convenience yield?

A convenience yield is a non-cash benefit of holding the physical commodity rather than a contract on it. It appears when inventories are low: a refinery that cannot afford to run out of crude will pay a premium for oil in hand, so spot prices are bid up and forward prices do not rise by the full cost of carry.

It works like income, so it enters the formula with the benefits. A forward price well below what interest and storage alone imply can be a sign of a large convenience yield.

Forward price against spot

Opportunity cost + other costs, against benefitsthe net cost of carry decides F₀(T) − S₀

Costs above benefits Memorize F₀(T) > S₀
Costs below benefits Memorize F₀(T) < S₀
Costs equal benefits Use it F₀(T) = S₀

Go deeper Why a forward below spot is normal for a high-yield index

If an index pays a dividend yield above the risk-free rate, i > r, then r − i is negative and F₀(T) < S₀. Nothing is wrong: the forward buyer simply gives up more income than the interest saved by waiting. Commodity markets with forward prices below spot are said to be in backwardation, and those with forward prices above spot in contango. The cost-of-carry formula explains both.

Go deeper Bonds: coupons are the income

For a bond, the coupons in the life of the forward are the income. Subtract their present value from the spot price before compounding. A par bond with a coupon above the risk-free rate has a forward price below spot. The forward module develops this.

Go deeper Why expected returns never appear in derivative prices

The replication argument prices a forward from the spot price and the loan, whatever the asset’s risk. The risk premium sits in the spot price already. This is the first sight of a theme that returns in option pricing, where the valuation uses risk-free growth and still gives the right answer.

Memorize this

MemorizeThe carry formulas

Inputs	Forward price
Known amounts $I$ , $C$ (discrete)	$F_0(T) = [S_0 - PV_0(I) + PV_0(C)](1+r)^T$
Rates $i$ , $c$ (continuous)	$F_0(T) = S_0\,e^{(r+c-i)T}$

Memory hook Costs go in with a plus, benefits with a minus. "Pay to hold, price goes up; get paid to hold, price goes down".

Know it cold Subtract the present value of income, not the amount. Convenience yield is a benefit (it enters like $i$ ). The risk premium never enters a forward price.

MemorizeForward against spot

Net of costs and benefits	Forward against spot
$r$ + costs > benefits	$F_0(T) > S_0$
$r$ + costs < benefits	$F_0(T) < S_0$
$r$ + costs = benefits	$F_0(T) = S_0$

Memory hook A positive r alone puts F above S. Income or convenience yield must beat r plus costs to pull F below S.

Know it cold A bond or index with income above the risk-free rate has $F_0(T)<S_0$ . A commodity with storage costs and no convenience yield has $F_0(T)>S_0$ .

MemorizeSpot against expected future price

Memory hook Two discount rates: spot uses a risk-adjusted rate, the forward uses the risk-free rate.

Know it cold $S_0 = E(S_T)/(1+r+\lambda)^T$ and $F_0(T)=S_0(1+r)^T$ . With a positive risk premium, $F_0(T) < E(S_T)$ . The forward price is not the expected spot price.

What goes into the cost of carry

Cost of carrycosts and benefits of holding the asset until T

Costs Memorize push the forward price up
- Opportunity cost, r Memorize interest forgone; applies to every asset
- Storage, insurance, transport, spoilage Memorize physical commodities
Benefits Memorize pull the forward price down
- Cash income Memorize dividends, coupons, index dividend yield
- Convenience yield Memorize non-cash benefit of holding the physical commodity

Cost of carry by asset class

Asset classwhat offsets the risk-free rate

Equities no storage cost
- Non-dividend stock none; only r
- Dividend-paying stock dividends lower F₀(T)
- Equity index dividend yield, a rate
Commodities physical holding
- Storage and insurance Memorize raise F₀(T)
- Convenience yield Memorize lowers F₀(T)
Bonds coupon income lowers F₀(T)
Foreign exchange rate differential r_f − r_d (next lesson)

Forward price against spot

Opportunity cost + other costs, against benefitsthe net cost of carry decides F₀(T) − S₀

Costs above benefits Memorize F₀(T) > S₀
Costs below benefits Memorize F₀(T) < S₀
Costs equal benefits Use it F₀(T) = S₀

Spot price from the expected future price

S_0 = \dfrac{E(S_T)}{(1 + r + \lambda)^T}

Quick example

An asset is expected to be worth 121 in two years and investors require 10% (r plus the risk premium λ). 121 ÷ 1.10^2 = 100.00. The forward uses r only.

On your calculator

1.1y^x2=1/x×121= 100

y^x power · 1/x reciprocal

121ENTER1.1ENTER2y^x÷ 100

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with known income and costs

F_0(T) = [S_0 - PV_0(I) + PV_0(C)]\,(1 + r)^T

Quick example

Share at 120.00, present value of dividends 3.50, r = 4%, six months. (120.00 − 3.50) × 1.04^0.5 = 118.81, against 122.38 with no dividends.

On your calculator

1.04y^x0.5=×(120−3.5)= 118.8072

y^x power

120ENTER3.5−1.04ENTER0.5y^x× 118.8072

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with yields and costs as rates

F_0(T) = S_0\,e^{(r + c - i)T}

Quick example

Index at 1,200, r = 3.2%, dividend yield 2.0%, three months (continuous). 1,200 × e^((0.032 − 0.02) × 0.25) = 1,203.61.

On your calculator

0.032−0.02=×0.25=2NDe^x×1200= 1203.605

2ND second function (the shifted label above a key) · e^x e to the power x

0.032ENTER0.02−0.25×ge^x1200× 1203.605

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

How it is asked

Forward with a dividend

Share 140, dividend 4 in three months, r = 5%, one year. Forward?

PV(I) = 4 ÷ 1.05^0.25 = 3.95; (140 − 3.95) × 1.05 = 142.85. Traps: 142.80, 143.00.
Equity index (continuous)

Index 2,400, r = 5%, yield 3%, one year. Forward?

2,400 × e^(0.05 − 0.03) = 2,448.48. Traps: 2,523.05 (yield ignored), 2,599.89 (yield added).
Sign of F₀(T) − S₀

Bond with a coupon above the risk-free rate; or a commodity with storage costs.

Coupon above r: F₀(T) < S₀. Storage costs and no yield: F₀(T) > S₀.
Forward against expected spot

Is the forward price the market's forecast of the spot price?

No. The forward compounds spot at r; spot discounts E(S_T) at a rate that includes the risk premium.

Exam traps

Subtracting the dividend itself instead of its present value, or compounding it wrongly. Discount each dividend to today at r, subtract from S₀, then compound.
Adding storage costs after compounding. Put the present value of the costs in the bracket so they compound too (or add the future value of the costs at T).
Saying the forward price equals the expected spot price. The forward compounds spot at the risk-free rate only; it has no risk premium in it.
Treating convenience yield as a cost. It is a benefit of holding the physical commodity, so it lowers the forward price.

Checkpoint

6 exam-style questions. About 90 seconds each.

1A commodity trades at USD 80. The present value of storage costs for one year is USD 2 and the risk-free rate is 5%. The one-year no-arbitrage forward price is closest to:
2An equity index is at 3,000. The risk-free rate is 3% and the dividend yield is 2% (both continuously compounded). The one-year forward price is closest to:
3A convenience yield on a commodity most likely:
4A share trades at USD 100 and will pay a dividend of USD 6 in three months. The risk-free rate is 8% and the contract expires in one year. The no-arbitrage forward price is closest to:
5An asset is expected to be worth USD 132 in one year. Investors require a 12% return, and the risk-free rate is 4%. The one-year forward price is closest to:
6A fixed-coupon bond is priced at par and its coupon rate is above the risk-free rate. Compared with the spot price, the forward price is most likely:

Key takeaways

Spot = expected future price discounted with a risk premium. Forward = spot compounded at the risk-free rate.
F₀(T) = [S₀ − PV(I) + PV(C)](1 + r)^T, or S₀e^((r + c − i)T).
Costs raise the forward price. Income and convenience yield lower it.
F₀(T) is above S₀ when r plus costs exceed benefits, and below when benefits are larger.

Spot price from the expected future price

S_0 = \dfrac{E(S_T)}{(1 + r + \lambda)^T}

Quick example

An asset is expected to be worth 121 in two years and investors require 10% (r plus the risk premium λ). 121 ÷ 1.10^2 = 100.00. The forward uses r only.

On your calculator

1.1y^x2=1/x×121= 100

y^x power · 1/x reciprocal

121ENTER1.1ENTER2y^x÷ 100

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with known income and costs

F_0(T) = [S_0 - PV_0(I) + PV_0(C)]\,(1 + r)^T

Quick example

Share at 120.00, present value of dividends 3.50, r = 4%, six months. (120.00 − 3.50) × 1.04^0.5 = 118.81, against 122.38 with no dividends.

On your calculator

1.04y^x0.5=×(120−3.5)= 118.8072

y^x power

120ENTER3.5−1.04ENTER0.5y^x× 118.8072

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with yields and costs as rates

F_0(T) = S_0\,e^{(r + c - i)T}

Quick example

Index at 1,200, r = 3.2%, dividend yield 2.0%, three months (continuous). 1,200 × e^((0.032 − 0.02) × 0.25) = 1,203.61.

On your calculator

0.032−0.02=×0.25=2NDe^x×1200= 1203.605

2ND second function (the shifted label above a key) · e^x e to the power x

0.032ENTER0.02−0.25×ge^x1200× 1203.605

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

All Derivatives formulas, every lesson

Practice exam-style questions on Arbitrage, Replication and the Cost of Carry

Next lesson Currency forward prices and the interest rate differential

Questions about this lesson?

Where this comes back

Currency forward prices and the interest rate differential Arbitrage, Replication and the Cost of Carry
Pricing and Valuing Forward Contracts and FRAs Derivatives
Pricing and Valuing Futures Contracts Derivatives
Natural Resources: Land, Timber, Farmland and Commodities Alternative Investments

Questions candidates ask

What is the cost of carry in derivatives?

The cost of carry is the net of the costs and benefits of holding the underlying until the contract ends. Interest forgone and storage are costs; dividends, coupons and convenience yield are benefits. Costs push the forward price above spot and benefits pull it below.

Is the forward price the expected future spot price?

No. The forward price is spot compounded at the risk-free rate. Spot itself is the expected future price discounted at a rate that includes a risk premium. For a risky asset with a positive premium, the forward price is below the expected spot price.

What is a convenience yield?

A convenience yield is a non-cash benefit of holding a physical commodity, which arises when inventories are low. It works like income: it lowers the forward price and can push it below spot.

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