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FRM Exam Part I · Commodity Forwards and Futures

Convenience Yield Explained for FRM Part I

Updated 11 October 2026 · Fact-checked

Convenience yield is the benefit of holding physical inventory that a holder of a futures contract does not get. It lowers the forward price: F = S × e^((r + u − y)T). Rearranged, y = r + u − ln(F ÷ S) ÷ T. A high yield produces backwardation.

Understand Convenience Yield

A forward or futures holder owns a claim on the commodity, not the commodity itself. Someone who holds the physical barrel, tonne or bushel can keep a refinery running, avoid a stock-out or sell into a sudden price spike. That benefit is the convenience yield. It accrues only to the owner of the physical asset.

The basic cost of carry model says the forward price equals the spot price grown at the financing cost plus storage cost. For commodities, storage cost is a real expense of holding inventory. Convenience yield works in the opposite direction. It is a benefit of holding inventory, so it reduces the net cost of carry.

The convenience yield is not directly observed. You back it out from market prices. If the observed forward price is lower than the pure cost-of-carry price (spot plus interest plus storage), the gap is explained by the convenience yield. So it is best seen as the yield that makes the cost-of-carry formula match the market.

When the convenience yield is larger than interest plus storage costs, the net cost of carry is negative. The forward price is then below spot and the curve slopes down. This is backwardation. When convenience yield is small, as in times of plentiful inventory, the forward price is above spot. This is contango. Low inventories usually mean a high convenience yield.

Convenience yield and storage cost are different things. Storage cost is a cash outflow paid by the holder. Convenience yield is an implied benefit that reduces the carry. Both are measured per year, and they enter the formula with opposite signs.

Key formulas to remember

Forward price with continuous compounding
F = S × e^((r + u − y)T)
r = risk-free rate, u = storage cost as a proportion of spot per year, y = convenience yield, T in years. All rates continuously compounded.
Implied convenience yield
y = r + u − ln(F ÷ S) ÷ T
Use the market F and S. Gives an annualised, continuously compounded yield.
Net cost of carry
c = r + u − y
F = S × e^(cT). If c < 0, the market is in backwardation.
Storage cost as a dollar amount
F = (S + U) × e^(rT) × e^(−yT)
Here U is the present value of storage costs. Use when storage is given in currency, not as a rate.
Bounds without convenience yield
F ≤ S × e^((r + u)T)
With convenience yield, F can fall well below S × e^(rT). Arbitrage gives only this upper bound.

How to solve Convenience Yield questions

Most questions give you spot, forward, r and u and ask for y, or give y and ask for F. Follow the same path each time.

  1. 1Write down S, F, T, r and u. Convert T to years.
  2. 2Check that all rates are continuously compounded. If a rate is annual compounding, convert it first with r = ln(1 + R).
  3. 3Check whether storage is given as a rate (u) or as a currency amount (U).
  4. 4Write F = S × e^((r + u − y)T) and take natural logs: ln(F ÷ S) = (r + u − y)T.
  5. 5Solve for the unknown: y = r + u − ln(F ÷ S) ÷ T, or F directly if y is known.
  6. 6Round only at the end and check the sign. A forward below spot should give a large y.
  7. 7Interpret the answer: y > r + u means backwardation, y < r + u means contango.

Quickest way: Log-ratio shortcut

When to use it: Use it when you are asked for the implied convenience yield from two prices and a time period.

  1. Compute ln(F ÷ S) ÷ T. This is the market net cost of carry.
  2. Subtract it from r + u. The result is y.
  3. If F < S, the market carry is negative, so y must exceed r + u.
  4. Eliminate options with the wrong sign before you finish the arithmetic.

Common mistakes in Convenience Yield

  • Adding convenience yield to the cost of carry

    Students see it as another cost, like storage.

    Fix: It is a benefit. It enters with a minus sign: r + u − y.

  • Treating convenience yield as an observable cash flow

    Storage costs are real payments, so the two look alike.

    Fix: Convenience yield is implied from prices. It is a plug that makes the cost-of-carry formula fit.

  • Using ln(F ÷ S) without dividing by T

    Students rush the algebra when T is not one year.

    Fix: Always divide by T. A six-month problem needs ln(F ÷ S) ÷ 0.5.

  • Mixing annual and continuous compounding

    Rates quoted as 5% are used directly in the exponential formula.

    Fix: Check the wording. Use the rate as continuous only if the question says so, otherwise convert.

  • Saying backwardation means the futures price is low because the commodity is cheap

    Students link the curve shape to value instead of carry.

    Fix: Backwardation means the convenience yield exceeds interest plus storage. It signals scarce inventory, not low price.

Worked examples

Example 1

Crude oil spot is $80.00. The one-year forward price is $76.00. The continuously compounded risk-free rate is 4% and storage costs are 3% of spot per year. What is the implied convenience yield?

Show the solution
  1. S = 80, F = 76, T = 1, r = 0.04, u = 0.03.
  2. ln(F ÷ S) = ln(76 ÷ 80) = ln(0.95) = −0.051293.
  3. Divide by T = 1: −0.051293.
  4. y = r + u − (−0.051293) = 0.07 + 0.051293 = 0.121293.

Answer: Implied convenience yield ≈ 12.13% per year. The market is in backwardation.

Example 2

Copper spot is $9,000 per tonne. The risk-free rate is 3%, storage costs are 2% of spot and the convenience yield is 6%, all continuously compounded per year. What is the fair six-month forward price?

Show the solution
  1. S = 9,000, T = 0.5.
  2. Net carry c = r + u − y = 0.03 + 0.02 − 0.06 = −0.01.
  3. Exponent = −0.01 × 0.5 = −0.005.
  4. e^(−0.005) ≈ 0.995012.
  5. F = 9,000 × 0.995012 = 8,955.11.

Answer: F ≈ $8,955 per tonne, below spot, so the curve is in backwardation.

Exam tips

  • Check the sign of your answer. If F is below S, the convenience yield must exceed r + u.
  • Read whether storage cost is a percentage or an amount. The formula changes with it.
  • Expect conceptual questions that link inventory levels to the curve. Low inventory means high convenience yield and backwardation.
  • A financial calculator's ln and e^x keys speed up the arithmetic. Keep at least six decimals in the intermediate step.

Practice questions from Commodity Forwards and Futures

Convenience Yield: frequently asked questions

What is convenience yield in simple terms?

It is the benefit of owning the physical commodity rather than a futures contract on it. Examples are avoiding a shortage and keeping production running. It cannot be seen directly and is implied from prices.

How do you calculate convenience yield from futures prices?

Use y = r + u − ln(F ÷ S) ÷ T with continuously compounded rates. Put in spot, forward, time in years, the risk-free rate and the storage cost rate. The result is an annual yield.

How is convenience yield related to backwardation?

When convenience yield is greater than r + u, the net cost of carry is negative. The forward price is then below spot and the market is in backwardation. Scarce inventory usually causes this.

What is the difference between convenience yield and storage cost?

Storage cost is a real expense of holding the commodity and it raises the forward price. Convenience yield is an implied benefit of holding it and it lowers the forward price.