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Financial Instruments, Markets and Conventions / Forwards, Futures, Swaps, and Contracts for Difference

Forward and Futures Prices: Cost-of-Carry, Known Income, and Continuous Yields

Every futures price used in the hedging examples of previous lessons — $79/barrel for crude oil, 1,010 for the S&P 500, an AUD forward — rests on a no-arbitrage equation that pins the futures price to the spot price. A barrel of oil available today and a futures contract to receive a barrel of oil in three months are not the same thing. The futures contract is different from the spot in ways that can be precisely quantified: the buyer of the futures does not pay interest on a loan (the money stays invested), but also does not receive any income (dividends, coupons, lease income) that the underlying asset pays during the life of the contract. The cost of carry captures all these effects in one number.

This note derives the no-arbitrage forward price for five cases — no income, known cash income, known continuous yield, currencies (covered interest rate parity), and commodities (with storage costs and convenience yield). The derivation in each case is the same two-strategy argument: two positions that deliver the same cash flow at time TT must cost the same at time 00.

Learning objectives. After this note you should be able to:
  • derive the forward price for an investment asset with no income, known income, and known yield;
  • compute the MTM value of a forward contract after initiation;
  • explain why FX forward prices obey covered interest rate parity as a special case of the cost-of-carry formula;
  • describe why the cost-of-carry formula is an inequality for consumption commodities, and define convenience yield;
  • state the unified cost-of-carry formula and apply it across all asset classes;
  • explain the relationship between futures prices and expected future spot prices.

In this lesson

  1. Investment assets vs consumption assets
  2. Case 1: No income
  3. Case 2: Known cash income
  4. Case 3: Known continuous yield
  5. Valuing a forward contract after initiation
  6. Stock index futures
  7. FX forward prices and covered interest rate parity
  8. Commodity futures
  9. The unified cost-of-carry formula
  10. Futures price vs expected future spot price

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