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FRM Exam Part I · Pricing Financial Forwards and Futures

Storage Costs, Convenience Yield and Commodity Forwards

Updated 11 October 2026 · Fact-checked

A commodity forward price equals the spot price grown at the risk-free rate plus storage costs, less the convenience yield: F = S × e^((r + u − y)T). Convenience yield is the benefit of holding the physical good. If it is high, the curve slopes down (backwardation).

Understand Storage Costs, Convenience Yield and Commodity Forwards

A commodity is different from a financial asset. You can store it, but storing costs money: warehouse rent, insurance, spoilage. You can also use it. A refinery holding crude oil can keep operating when supply is tight. That benefit of holding the physical good is the convenience yield.

Start with the cost-of-carry idea. If you buy the commodity now and sell it forward, you pay the spot price, fund it at the risk-free rate, and pay storage. So the no-arbitrage forward price is the spot price carried forward at the net cost of carry. Storage costs push the forward price up. Convenience yield pushes it down, because holders of the physical good get a benefit that a forward holder does not.

Storage costs can be treated two ways. As a continuous proportional cost u (like a negative yield), the forward is F = S × e^((r + u − y)T). As a known dollar amount, you add the present value of the storage costs to the spot price: F = (S + PV of storage) × e^(rT). Both ideas say the same thing: storage is a cost of carry.

Convenience yield is not directly observable. You infer it from market prices. For a commodity held for consumption, the no-arbitrage relation is an inequality: F ≤ S × e^((r + u)T). If F were above this bound, you could buy the commodity, sell the forward and earn an arbitrage profit. The reverse trade, selling the commodity and buying the forward, does not work as an arbitrage. Holders of the physical commodity value the convenience of holding it, so they will not sell it to capture the arbitrage. The convenience yield y is the value that turns the inequality into an equality.

Contango means the futures price is above spot and the curve slopes up with maturity. Backwardation means the futures price is below spot and the curve slopes down. Roughly, contango appears when r + u > y, and backwardation when y > r + u. Treat this as a description of carry, not a forecast of the future spot price.

Key formulas to remember

Forward with proportional storage cost and convenience yield (continuous)
F = S × e^((r + u − y)T)
r = risk-free rate, u = storage cost as a yearly rate, y = convenience yield, all continuously compounded.
Forward with known dollar storage costs
F = (S + U) × e^(rT)
U = present value of all storage costs over the life, discounted at r. Storage paid at the end of the period needs discounting first.
Cost-of-carry upper bound
F ≤ S × e^((r + u)T)
If F exceeds this, buy the commodity, sell the forward and earn arbitrage. The reverse arbitrage (sell the commodity, buy the forward) fails because holders of the physical commodity value the convenience of holding it and will not sell it to capture the arbitrage.
Implied convenience yield
y = r + u − ln(F ÷ S) ÷ T
Use continuous compounding. Rearranged from the pricing formula.
Curve shape rule
Contango: F > S. Backwardation: F < S.
Contango roughly when r + u > y. Backwardation roughly when y > r + u.

How to solve Storage Costs, Convenience Yield and Commodity Forwards questions

Use this order for any commodity forward question.

  1. 1Identify what is given: spot S, maturity T in years, risk-free rate r, storage cost (rate or dollar amount), and convenience yield or forward price.
  2. 2Check the compounding. The formulas above assume continuous compounding. Convert annual rates if the question says so.
  3. 3Decide how storage is stated. If it is a yearly rate, add it to r in the exponent. If it is a dollar amount, add its present value to S.
  4. 4Subtract the convenience yield in the exponent when it is given.
  5. 5Compute the forward price, or rearrange to solve for y using y = r + u − ln(F ÷ S) ÷ T.
  6. 6Compare F with S to name contango or backwardation.
  7. 7Sanity check: storage should raise F, convenience yield should lower F.

Quickest way: Net carry shortcut

When to use it: Use when the question gives rates and asks for a forward price or convenience yield, and the answer options are spaced apart.

  1. Add r and u, then subtract y to get the net carry c.
  2. Compute F = S × e^(cT) with a calculator's e^x key.
  3. To find y, compute ln(F ÷ S) ÷ T first, then y = r + u minus that number.
  4. If c is positive the curve is in contango, if negative in backwardation. Use this to eliminate wrong options fast.

Common mistakes in Storage Costs, Convenience Yield and Commodity Forwards

  • Adding the convenience yield instead of subtracting it.

    Students treat it like a cost because it sounds like extra value attached to the commodity.

    Fix: Remember it is a benefit of holding the physical good, so it works like a dividend yield and reduces the forward price.

  • Forgetting to discount a dollar storage cost paid later.

    Storage costs are listed as a simple amount and are added directly to spot.

    Fix: Add the present value of storage costs to S, then compound the sum at r.

  • Mixing simple and continuous rates.

    The question gives an annual rate and the formula uses e^(rT).

    Fix: Read the compounding statement. Use the exponent form only with continuous rates, or convert first.

  • Saying backwardation means the spot price will fall (or contango means it will rise).

    Students read curve shape as a forecast.

    Fix: The curve shape reflects carry costs and convenience yield. It is not a guaranteed prediction of the spot price.

  • Claiming F = S × e^((r + u)T) always holds as an equality for commodities.

    Students carry over the financial asset formula.

    Fix: For consumption commodities it is an upper bound. Convenience yield closes the gap.

  • Using months instead of years for T.

    The maturity is stated in months, such as 9 months.

    Fix: Convert to years before using the formula: 9 months = 0.75.

Worked examples

Example 1

Spot copper is $8,000 per tonne. The risk-free rate is 4% and storage costs are 2% of the spot price per year, both continuously compounded. The convenience yield is 1%. What is the 1-year forward price?

Show the solution
  1. Net carry = r + u − y = 4% + 2% − 1% = 5%.
  2. F = 8,000 × e^(0.05 × 1).
  3. e^0.05 = 1.051271.
  4. F = 8,000 × 1.051271 = 8,410.17.

Answer: The 1-year forward price is about $8,410 per tonne (contango, since F > S).

Example 2

Crude oil spot is $80 per barrel. The 6-month futures price is $78. The risk-free rate is 5% and storage costs are 3% per year, both continuously compounded. What is the implied convenience yield?

Show the solution
  1. T = 0.5 years.
  2. ln(F ÷ S) = ln(78 ÷ 80) = ln(0.975) = −0.025318.
  3. Divide by T: −0.025318 ÷ 0.5 = −0.050636.
  4. y = r + u − (−0.050636) = 0.05 + 0.03 + 0.050636 = 0.130636.
  5. Check: F = 80 × e^((0.08 − 0.130636) × 0.5) = 80 × e^(−0.025318) = 78. Correct.

Answer: The implied convenience yield is about 13.06% per year. F < S, so the market is in backwardation.

Exam tips

  • Know the sign of each term: r and u raise the forward, y lowers it. Many wrong options differ only by a sign.
  • Check whether storage is a rate or a dollar amount. They are handled differently.
  • When asked about contango versus backwardation, link the shape to r + u compared with y.
  • Expect a conceptual question on why convenience yield exists and why the cost-of-carry relation is only an inequality for consumption commodities.

Practice questions from Pricing Financial Forwards and Futures

Storage Costs, Convenience Yield and Commodity Forwards: frequently asked questions

What is convenience yield in simple terms?

It is the benefit of holding the physical commodity rather than a futures contract on it. Examples are keeping a production line running or avoiding stockouts. It is high when supplies are tight.

What is the difference between contango and backwardation?

In contango, futures prices are above spot and rise with maturity. In backwardation, futures prices are below spot and fall with maturity. Contango tends to appear when carry costs exceed convenience yield, and backwardation when convenience yield is larger.

How do you calculate convenience yield from futures prices?

Use y = r + u − ln(F ÷ S) ÷ T with continuous compounding. Put in the spot, futures price, maturity in years, risk-free rate and storage cost rate.

Can convenience yield be negative?

For a commodity held for consumption, no-arbitrage implies y ≥ 0, because F cannot exceed S × e^((r + u)T). A negative computed value would mean F is above that bound, which signals a cash-and-carry arbitrage. Before concluding that, check your inputs and compounding for a mistake.