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Financial Instruments, Markets and Conventions / Commodities and the Cost-of-Carry World

Convenience Yield, Storage Costs, and the Theory of Storage

A copper fabricator holding 1,000 tonnes of copper in a warehouse is willing to accept a futures price below (S0+U)erT(S_0 + U) e^{rT}, even though that would look like a pricing anomaly to a financial-asset trader. The reason: the physical copper can run the plant tomorrow. A long futures position cannot. This shadow value of inventory — the convenience yield — is the gap between what the cost-of-carry formula predicts and what the market quotes for a consumption commodity.

The theory of storage, developed by Working (1948–49) and Brennan (1958), formalises this intuition: the convenience yield is a decreasing function of inventory. When warehouses are full, the marginal barrel of oil adds nothing to supply security — convenience yield is low or zero and the forward curve slopes upward (contango). When inventories are tight, the next barrel is precious — convenience yield is high and the forward curve slopes downward (backwardation). This lesson derives the pricing equations and maps them to the observable shape of the forward curve.

Learning objectives. After this note you should be able to:
  • state the cost-of-carry pricing equation for investment commodities with storage costs and explain why equality holds;
  • explain why only an inequality holds for consumption commodities and derive the convenience yield from the observed futures price;
  • write the unified cost-of-carry formula and identify all components;
  • connect inventory levels to the magnitude of the convenience yield;
  • determine the optimal delivery timing under a futures contract given the shape of the forward curve.

In this lesson

  1. Investment commodities with storage costs
  2. Consumption commodities and the pricing inequality
  3. The convenience yield
  4. The cost-of-carry: a unified formula
  5. Delivery options and timing implications

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