How do storage costs and convenience yield determine whether a commodity futures curve is in contango or backwardation?

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AcadiFi · Accepted Answer

The shape of a commodity futures curve depends on the tug-of-war between **costs of carry** (storage + financing) and the **convenience yield** from holding physical inventory. **Full Cost-of-Carry Model** **F = S x e^{(r + u - y) x T}** where: - **r** = risk-free rate - **u** = storage cost rate (as a continuous yield) - **y** = convenience yield (benefit of holding physical) **When F > S (contango):** r + u > y — carrying costs dominate. This is typical for storable, non-perishable commodities with ample inventories (gold, copper in surplus periods). **When F < S (backwardation):** y > r + u — the convenience yield dominates. This happens when physical inventories are tight and immediate delivery is valuable (crude oil during supply disruptions, soybeans during drought). **Worked Example** Hartwell Refining is analyzing crude oil. Spot WTI is $78.40/barrel. The 6-month futures contract trades at $76.90. The risk-free rate is 4.8% and storage costs are approximately 2.1% annualized. Implied convenience yield: 76.90 = 78.40 x e^{(0.048 + 0.021 - y) x 0.5} ln(76.90 / 78.40) = (0.069 - y) x 0.5 -0.01935 = 0.0345 - 0.5y y = (0.0345 + 0.01935) / 0.5 = **10.77%** A convenience yield of 10.77% implies the market places a high premium on having physical barrels available — likely due to refinery maintenance schedules or supply concerns. ```mermaid flowchart TD A["Spot Price: $78.40"] --> B{"r + u vs y?"} B -->|"r + u = 6.9% < y = 10.77%"| C["Backwardation: F < S"] B -->|"r + u > y"| D["Contango: F > S"] C --> E["Physical holders earn convenience yield"] D --> F["Futures holders earn roll yield"] ``` **Why Hold Physical in Backwardation?** Refineries, utilities, and manufacturers need physical feedstock to keep operations running. Running out of crude oil shuts down a refinery — that operational risk far exceeds any financial gain from selling spot and buying cheaper futures. The convenience yield quantifies this operational insurance. For more on commodity derivatives, visit our FRM Part I course.

How do storage costs and convenience yield determine whether a commodity futures curve is in contango or backwardation?

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