Lesson 2 of 3 6–9% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Commodities: sectors, ways to invest, contango and backwardation

Q: What is the difference between contango and backwardation?

In contango, futures prices are above the spot price because carrying costs exceed the convenience yield, so the curve slopes up. In backwardation, futures are below spot because the convenience yield exceeds carrying costs, so the curve slopes down. Low inventories favor backwardation.

Q: What is convenience yield?

The non-cash benefit of holding the physical commodity rather than a futures contract, such as guaranteed supply when stocks are low. It is higher when inventories are low, accrues to the owner, and lowers the futures price in the pricing formula.

Q: How do investors invest in commodities?

Mostly through futures and other derivatives, which are liquid and aid price discovery. Alternatives are exchange-traded products, commodity trading advisers and managed accounts, specialized funds, producer shares and, rarely, the physical good.

Commodities produce no cash flows and cost money to store, so most investors use futures rather than physical goods. Futures prices equal spot grown by interest and storage costs less any convenience yield. When futures sit above spot the market is in contango; below spot, backwardation.

8 min cram, 32 min full read. Based on the 2027 CFA® Program Level I curriculum, Alternative Investments, Learning Module 5. 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: Raw land, timberland and farmland as investments, Arbitrage, Replication and the Cost of Carry

On the exam

Describe features of commodities and their investment characteristics

Describe the features of commodities and their investment characteristics. Expect: the forms of investment; the futures pricing formula with a convenience yield; contango versus backwardation; which way low inventories push the curve.

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.

What are commodities?

Commodity sectors group the basic goods that feed the economy: energy, base metals, precious metals, agriculture and a catch-all “other”. Some are “hard” (mined or extracted, like copper and oil) and some “soft” (grown, like grains and coffee), which are the hard and soft commodities labels.

Commodity sectors

Commoditiesno cash flows; returns come from price changes beyond the cost of carry

Energy Memorize crude oil (Brent, West Texas Intermediate or WTI), natural gas, electricity, coal
Base metals Memorize copper, aluminum, zinc, lead, tin, nickel
Precious metals Memorize gold, silver, platinum, palladium
Agriculture Memorize grains, livestock, coffee, sugar
Other Memorize carbon credits, freight, forest products

A commodity does not generate cash flows, and holding it costs money for transport, storage and insurance. The cost of carry is the full cost of holding the physical good, including the interest forgone on the money tied up in it. Investors therefore earn only if the price rises by more than the cost of carry, because they are betting on future economic value rather than on using the good.

Governments matter more here than in most asset classes. They subsidize food and support farmers, control rights to extract oil, gas and minerals, and own large producers in many emerging markets. Climate policy also reshapes demand, for example by raising demand for the metals used in batteries. Commodities also come in many grades and delivery locations, so derivative contracts specify quantity, quality, maturity and delivery location.

How do investors get commodity exposure?

Most commodity investing is done through derivatives. Holding the physical good brings storage, insurance, transport and brokerage costs, tax complications and little price transparency.

Commodity futures (with forwards, and options on futures) are the main route. A future is an obligation to buy or sell a set amount at a fixed price, place and date. Futures are exchange traded, marked to market daily, liquid and good for price discovery, and counterparty risk is handled by the clearinghouse. The delivery obligation matters in a crisis: in extreme storage shortages, futures prices have even fallen below zero, as when oil storage ran out in 2020.

Ways to get commodity exposure

Commodity exposuremost of it is taken through derivatives

Derivatives (the main route) Memorize futures, forwards, options on futures; liquid, exchange traded, price discovery
Exchange-traded products Use it ETFs and ETNs for investors limited to securities or wanting a simple brokerage account; some leveraged or inverse
CTAs and SMAs Use it managed futures: directional trades in futures; an SMA is a tailored account
Specialized funds Know the idea sector expertise, e.g. private energy partnerships or energy mutual funds
Producer shares and physical Know the idea indirect, adds financing and operating risk; or hold the physical good and bear storage costs

Exchange-traded products (ETPs) are exchange-traded funds (ETFs) or exchange-traded notes (ETNs) that suit investors restricted to listed shares or wanting a simple brokerage account. They may hold physical metal or futures, can use leverage, can be inverse (“bearish”), and charge expense ratios.
Commodity trading advisers (CTAs) run managed futures strategies that take directional positions based on technical and fundamental signals. Modern CTAs often trade equities, bonds and currencies too, so a commodity-only investor must pick carefully. A separately managed account (SMA) is a tailored version for one investor (more in the hedge funds module).
Specialized commodity funds bring sector expertise: private energy partnerships have private-equity-style terms and are used by institutions, and public energy funds focus on oil and gas.
Producer shares are an indirect route. They add financing and operating risk, and, if privately held, illiquidity.

InteractiveWhat bends the futures curve

Try this 0 of 3

Raise the convenience yield until the curve turns downward
Make the curve perfectly flat
Set r 3.5%, c 1% and i 6%, then read the 12-month price

Aluminum trades at USD 2,500 a metric ton. Move the convenience yield and watch the curve tilt.

Curve shape

Risk-free rate r3.5%

Storage cost c1%

Convenience yield i0%

12-month futuresUSD 2,615.07

Net carry r + c − i4.5%

Curve shapeContango

Futures minus spot+115.07

Futures price

InventoriesPlentiful

ScarcePlentiful

Carry costs (4.5%) exceed the convenience yield (0%): futures sit above spot. That is contango, which tends to cost a long investor as contracts roll into dearer ones.

Convenience yield is a benefit of holding the physical good, so it lowers the futures price. It rises when inventories run low.

Show working

F₀(T) = S₀ × e^((r + c − i) × T), with T in years.

Net carry: 3.5% + 1% − 0% = 4.5%.

F₀(1) = 2,500 × e^(0.045) = 2,500 × 1.046028 = USD 2,615.07.

Six months: 2,500 × e^(0.045 × 0.5) = USD 2,556.89.

12-month futures price on your calculator

0.035+0.01=net carry 0.045 for one year
2NDe^x1.046028
×2500=2,615.07

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

0.035ENTER0.01+r + c
ge^x1.046028
2500×2,615.07

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

Display: F₀(1) = USD 2,615.07

Scarce inventory makes owning the physical worth more, pulling futures below spot.

How is a commodity forward priced?

Memorize

Because commodities trade in both physical and derivative markets, a no-arbitrage condition links the two prices. Before any benefit of holding the good, the forward must exceed spot by the cost of carry: the risk-free rate $r$ (opportunity cost) plus storage, transport and insurance $c$ . Holding the physical good can also bring a non-cash benefit, the convenience yield $i$ , which is the premium placed on having the good in hand. It is related inversely to inventories: the lower the stocks, the higher the convenience yield. Because it accrues to the owner, it reduces the forward price.

Commodity forward price with convenience yield Memorize

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

Worked example One-year aluminum forward, two convenience yields

Aluminum trades at USD 2,500 per metric ton. The risk-free rate is 3.5% and storage costs 1% a year (continuous), so $r + c = 4.5\%$ .

Convenience yield 0: F = 2,500 × e^0.045 = USD 2,615.07. Futures are above spot: contango.
Convenience yield 6%: F = 2,500 × e^(0.035 + 0.01 − 0.06) = 2,500 × e^(−0.015) = USD 2,462.78. Futures are below spot: backwardation.

The same spot price gives two curve shapes. Only the convenience yield changed.

Step 1 on your calculator

0.035+0.01=2NDe^x×2500= 2615.07

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

0.035ENTER0.01+ge^x2500× 2615.07

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

Step 2 on your calculator

0.035+0.01−0.06=2NDe^x×2500= 2462.78

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

0.035ENTER0.01+0.06−ge^x2500× 2462.78

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

Rearrange itCommodity forward price: solve for any input

The question asks for

A commodity has a spot price of 2,500.00. The risk-free rate is 3.50%, storage costs are 1.00% and the convenience yield is 6.00% (all continuously compounded). The 1-year forward price is closest to:

Spot price, S₀Risk-free rate, r%Storage and carry cost, c%Convenience yield, i%Time to expiry in years, T

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

2,500.00 × e^((0.035 + 0.01 − 0.06) × 1) = 2,462.78

Forward price, F₀(T)2,462.78

On your calculator

0.035+0.01−0.06=×1=2NDe^x×2500=2462.78

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

0.035ENTER0.01+0.06−1×ge^x2500×2462.78

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

What are contango and backwardation?

Memorize

The curve shape depends on which is larger, carry costs ( $r + c$ ) or the convenience yield ( $i$ ).

Curve shape rule Memorize

i < r + c \Rightarrow \text{contango};\quad i > r + c \Rightarrow \text{backwardation}

Contango and backwardation

Futures curve versus spotset by carry costs versus convenience yield

Contango Memorize futures above spot; upward curve; carry costs exceed convenience yield; inventories ample
- Effect on a long-only investor tends to lower returns
Backwardation Memorize futures below spot; inverted curve; convenience yield exceeds carry costs; inventories low
- Effect on a long-only investor tends to raise returns

Contango means the forward is above spot: the cost of ownership exceeds the benefit of the convenience yield, which is typical when inventories are ample. Backwardation means the forward is below spot, a downward-sloping or inverted curve, which arises when the benefit of holding the physical good exceeds the carry costs, typically when inventories are low.

The rule of thumb for a long-only investor: contango tends to lower returns, backwardation tends to raise them. Market conditions can flip the shape quickly. In early 2020 the oil curve was in backwardation with low stocks, then slid into steep contango when demand collapsed and storage filled.

Exam tip ”Contango = costs win, forward above spot. Backwardation = convenience wins, forward below spot.” Never describe backwardation as a forward above spot.

Go deeper Why contango erodes long-only returns (roll yield)

Index and ETP investors hold futures, not spot. Before a contract expires they sell it and buy the next one. In contango the next contract is dearer, so they sell low and buy high: the roll costs them even if spot is unchanged. In backwardation they sell dearer contracts and buy cheaper ones, which adds to the return. This is the intuition behind the rule of thumb; no roll formulas are needed at Level I.

Go deeper Negative futures prices

A futures contract is an obligation to take delivery. If storage is full and buyers disappear, holders who cannot take delivery pay others to take the contract, and the price can go below zero. The pricing formula assumes a positive price and a storable good, so it breaks down at that point.

Memorize this

MemorizeCommodity sectors

Sector	Examples
Energy	oil, natural gas, electricity, coal
Base metals	copper, aluminum, zinc, lead, tin, nickel
Precious metals	gold, silver, platinum, palladium
Agriculture	grains, livestock, coffee
Other	carbon credits, freight, forest products

Memory hook EBPAO: Energy, Base, Precious, Agriculture, Other.

Know it cold Platinum is a precious metal; nickel is base. Commodities have no cash flows and cost money to hold.

MemorizePricing formula and curve shapes

Memory hook Carry costs push the forward up (r and c); convenience yield pulls it down (i). Whichever is bigger wins.

Know it cold $F_0(T) = S_0 e^{(r + c - i)T}$ . Contango: $F > S$ ( $i < r + c$ ). Backwardation: $F < S$ ( $i > r + c$ ). Convenience yield is higher when inventories are low and is never a cost.

MemorizeWays to invest

Memory hook Futures first; everything else wraps futures or shares.

Know it cold Most exposure is through futures (liquid, price discovery, daily margin). ETPs suit investors limited to securities. CTAs trade directional futures strategies. Specialized funds bring sector expertise.

Commodity sectors

Commoditiesno cash flows; returns come from price changes beyond the cost of carry

Energy Memorize crude oil (Brent, West Texas Intermediate or WTI), natural gas, electricity, coal
Base metals Memorize copper, aluminum, zinc, lead, tin, nickel
Precious metals Memorize gold, silver, platinum, palladium
Agriculture Memorize grains, livestock, coffee, sugar
Other Memorize carbon credits, freight, forest products

Ways to get commodity exposure

Commodity exposuremost of it is taken through derivatives

Derivatives (the main route) Memorize futures, forwards, options on futures; liquid, exchange traded, price discovery
Exchange-traded products Use it ETFs and ETNs for investors limited to securities or wanting a simple brokerage account; some leveraged or inverse
CTAs and SMAs Use it managed futures: directional trades in futures; an SMA is a tailored account
Specialized funds Know the idea sector expertise, e.g. private energy partnerships or energy mutual funds
Producer shares and physical Know the idea indirect, adds financing and operating risk; or hold the physical good and bear storage costs

Contango and backwardation

Futures curve versus spotset by carry costs versus convenience yield

Contango Memorize futures above spot; upward curve; carry costs exceed convenience yield; inventories ample
- Effect on a long-only investor tends to lower returns
Backwardation Memorize futures below spot; inverted curve; convenience yield exceeds carry costs; inventories low
- Effect on a long-only investor tends to raise returns

Commodity forward price with convenience yield

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

Quick example

Spot 1,600, risk-free rate 3%, carry cost 1%, convenience yield 0.5%, two years. $1{,}600\,e^{(0.03 + 0.01 - 0.005)\times 2}$ = 1,600 × e^0.07 = 1,716.01.

On your calculator

0.03+0.01−0.005=×2=2NDe^x×1600= 1716.01

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

0.03ENTER0.01+0.005−2×ge^x1600× 1716.01

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

Implied convenience yield

i = r + c - \dfrac{1}{T}\ln\!\left(\dfrac{F_0}{S_0}\right)

Quick example

Spot 1,600, one-year futures 1,630, r = 3%, c = 1%. ln(1,630 ÷ 1,600) = 1.858%, so i = 4% − 1.86% = 2.14%. Futures above spot, so contango, because i is below r + c = 4%.

On your calculator

1630÷1600=LN+|-+0.03+0.01= 0.0214

LN natural logarithm · +|- change sign

0.03ENTER0.01+1630ENTER1600÷gLN− 0.0214

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Curve shape rule

i < r + c \Rightarrow \text{contango};\quad i > r + c \Rightarrow \text{backwardation}

Quick example

r = 3.5%, c = 1%, so the cut-off is 4.5%. A convenience yield of 3% gives contango; 6% gives backwardation; exactly 4.5% gives a flat curve.

How it is asked

Price with a convenience yield

Spot 120, r = 2.5%, storage 4.5%, convenience yield 1.5%, nine months. Futures price?

120 × e^((0.025 + 0.045 − 0.015) × 0.75) = 125.05. Traps: 120.90 (storage left out) and 126.47 (convenience yield left out).
Name the shape, or link it to inventories

Futures prices are below spot (or inventories are very low). Which curve shape?

Backwardation: a high convenience yield beats carry costs. Trap: contango (the reverse).
Which vehicle

Investor limited to listed securities wants commodity exposure. Which vehicle?

An exchange-traded product. Trap: a CTA (a managed futures strategy, not a simple listed security).
Effect on a long-only investor

The market moves into contango. Effect on a long-only futures investor?

Tends to lower returns (rolling into dearer contracts). Backwardation raises them.

Exam traps

Defining contango as forward below spot. Contango is forward above spot.
Treating convenience yield as a cost. It is a benefit of holding the physical good and lowers the forward price.
Saying most commodity investing is in physical goods. Most is through futures and other derivatives.
Leaving storage (c) out of the formula, or adding the convenience yield instead of subtracting it.
Assuming commodities produce income. They have no cash flows and carry costs.

Checkpoint

6 exam-style questions. About 90 seconds each.

1A commodity's futures prices are below its spot price. This is best described as:
2Spot oil is USD 80. The risk-free rate is 3%, storage costs are 5% and the convenience yield is 2% (all continuously compounded, annual). The six-month futures price is closest to:
3Most investors gain commodity exposure through:
4Copper trades at USD 9,000 per metric ton. A one-year futures contract trades at USD 9,300. The risk-free rate is 4% and storage costs are 2%, both continuously compounded. The implied convenience yield is closest to:
5Inventories of a commodity fall to very low levels. The futures curve is most likely to move toward:
6An investor who can hold only exchange-listed securities wants commodity exposure with simple trading through a brokerage account. The investor would most likely choose:

Key takeaways

Commodities: no cash flows, costs to hold. Sectors: energy, base metals, precious metals, agriculture, other.
Most exposure is through futures; ETPs, CTAs and SMAs, specialized funds and producer shares are alternatives.
$F_0(T) = S_0 e^{(r + c - i)T}$; convenience yield is a benefit and is higher when inventories are low.
Contango ($F > S$) tends to lower long-only returns; backwardation ($F < S$) tends to raise them.

Commodity forward price with convenience yield

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

Quick example

Spot 1,600, risk-free rate 3%, carry cost 1%, convenience yield 0.5%, two years. $1{,}600\,e^{(0.03 + 0.01 - 0.005)\times 2}$ = 1,600 × e^0.07 = 1,716.01.

On your calculator

0.03+0.01−0.005=×2=2NDe^x×1600= 1716.01

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

0.03ENTER0.01+0.005−2×ge^x1600× 1716.01

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

Implied convenience yield

i = r + c - \dfrac{1}{T}\ln\!\left(\dfrac{F_0}{S_0}\right)

Quick example

Spot 1,600, one-year futures 1,630, r = 3%, c = 1%. ln(1,630 ÷ 1,600) = 1.858%, so i = 4% − 1.86% = 2.14%. Futures above spot, so contango, because i is below r + c = 4%.

On your calculator

1630÷1600=LN+|-+0.03+0.01= 0.0214

LN natural logarithm · +|- change sign

0.03ENTER0.01+1630ENTER1600÷gLN− 0.0214

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Curve shape rule

i < r + c \Rightarrow \text{contango};\quad i > r + c \Rightarrow \text{backwardation}

Quick example

r = 3.5%, c = 1%, so the cut-off is 4.5%. A convenience yield of 3% gives contango; 6% gives backwardation; exactly 4.5% gives a flat curve.

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Practice exam-style questions on Natural Resources: Land, Timber, Farmland and Commodities

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Natural resources: risk, return, inflation hedging and diversification Natural Resources: Land, Timber, Farmland and Commodities
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Questions candidates ask

What is the difference between contango and backwardation?

In contango, futures prices are above the spot price because carrying costs exceed the convenience yield, so the curve slopes up. In backwardation, futures are below spot because the convenience yield exceeds carrying costs, so the curve slopes down. Low inventories favor backwardation.

What is convenience yield?

The non-cash benefit of holding the physical commodity rather than a futures contract, such as guaranteed supply when stocks are low. It is higher when inventories are low, accrues to the owner, and lowers the futures price in the pricing formula.

How do investors invest in commodities?

Mostly through futures and other derivatives, which are liquid and aid price discovery. Alternatives are exchange-traded products, commodity trading advisers and managed accounts, specialized funds, producer shares and, rarely, the physical good.

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