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Explain carrying costs and how they shift the forward price

CFA® Level I · Derivatives · Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives · DE.5.b

The idea

The forward price equals the spot price compounded at the risk-free rate, adjusted for anything the underlying costs or yields while it is held. Storage and insurance raise the forward price because the seller must be compensated for carrying the asset; dividends, coupons and convenience yield lower it because holding the spot asset confers a benefit the forward holder does not receive. Where benefits exceed financing and storage costs the forward price falls below spot, which is the condition usually called backwardation.

Formulas

Forward price with carry costs and benefits

F₀ = (S₀ + storage − income) × (1 + r)ᵗ

S₀
the spot price today
storage
the present value of the cost of holding the asset
income
the present value of what holding it pays — dividends, coupons, a convenience yield
r
the risk-free rate
t
the years to delivery

Costs of carry raise the forward price and benefits lower it, because the buyer of the forward avoids the one and gives up the other. Income large enough makes the forward cheaper than the spot.

A practice question

From the question bank, with the reason each wrong option is wrong.

How does a dividend paid on the underlying affect the forward price of a stock?

  1. It raises the forward priceWhy not: The dividend goes to whoever holds the shares, not to the forward buyer, so it reduces what the buyer must pay.
  2. It lowers the forward price — correct
  3. It has no effectWhy not: Dividends are part of the carry calculation and enter the forward price directly.

The forward holder does not receive the dividend, so the forward price is reduced by its present value to keep the two positions equivalent.

A harder one

Why does storage cost raise the forward price of a commodity?

  1. Because the holder must be compensated — correct
  2. Because storage is taxed as incomeWhy not: Storage is an expense rather than income, and tax treatment has no role in the no-arbitrage argument.
  3. Because exchanges add storage to the settlement priceWhy not: No exchange adds a storage line. The effect arises from the cost of carrying the physical commodity.

Whoever carries the asset until delivery bears a real cost. The forward price must cover financing plus storage, or the arbitrage of buying spot and selling forward would be unprofitable.

The hardest tier

A commodity's forward price is below its spot price. What does this imply about the balance of carry?

  1. Storage costs exceed the risk-free rateWhy not: Both are costs and both push the forward price up, so neither can explain it sitting below spot.
  2. The market expects the commodity to become worthlessWhy not: Forward prices are not expectations. Backwardation reflects a benefit to holding the physical asset now.
  3. The benefits of holding exceed the costs — correct

Convenience yield and the other benefits of physical possession outweigh financing and storage. A user who needs the commodity on hand values having it, and that convenience yield is a benefit the forward does not deliver; when it is large enough the forward price sits below spot.

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