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Set out the formula by which commodity forwards are priced.

FRM® Part I · Financial Markets and Products · Commodity Forwards and Futures · FRM1.FM.11.d

The idea

A commodity forward is priced by the cost of carry, with all rates continuously compounded and T in years: F0 = S0 × e^((r + λ − c) × T). In words, the forward is spot grown at the risk-free rate r, plus the storage cost rate λ, less the convenience yield c, the benefit that holders of the physical commodity enjoy. If instead the commodity can be lent out at a lease rate δ, the formula becomes F0 = S0 × e^((r − δ) × T). If storage costs are fixed cash amounts instead of a rate, grow spot at r and add the future value of the storage costs at delivery. Common mistakes are subtracting storage costs, adding the convenience yield, or forgetting that T must be in years. The formula assumes arbitrage works in both directions, which holds best for commodities that can be stored and lent.

Formulas

Cost-of-carry forward price of a commodity

F0 = S0 × e^((r + λ − c) × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
λ
the storage cost, as a continuous rate per year
c
the convenience yield, as a continuous rate per year
T
the time to delivery, in years

Storage is added and the convenience yield subtracted; reversing either sign is the usual mistake. Assumes arbitrage works both ways, which holds best for commodities that can be stored and lent.

Forward price of a commodity that can be lent

F0 = S0 × e^((r − δ) × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
δ
the lease rate, as a continuous rate per year
T
the time to delivery, in years

The lease rate works like a dividend yield, so a higher δ lowers the forward price.

A practice question

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

With continuously compounded rates, the no-arbitrage forward price of a commodity with storage cost rate lambda and convenience yield c is:

  1. F = S x exp((r - lambda + c) x T)Why not: Reverses the signs of both storage cost and convenience yield.
  2. F = S x exp((r + lambda + c) x T)Why not: Adds the convenience yield; it should be subtracted because it is a benefit of holding.
  3. F = S x exp((lambda - c) x T)Why not: Omits the risk-free rate, the financing cost of buying the commodity today.
  4. F = S x exp((r + lambda - c) x T) - correct

Cost of carry: interest and storage are costs of holding the commodity (added) and convenience yield is a benefit (subtracted), all compounded over T.

A harder one

A commodity trades at USD 95. The risk-free rate is 4%, storage costs are 2.5% and the convenience yield is 1.5%, all continuously compounded and per year. The nine-month forward price is closest to:

  1. USD 99.75Why not: Leaves out the convenience yield, giving full cost of carry.
  2. USD 97.89Why not: Uses the risk-free rate only, ignoring both storage costs and convenience yield.
  3. USD 98.63 - correct
  4. USD 95.00Why not: Subtracts storage costs along with the convenience yield.

Net carry = 0.050 per year; F = 95 x exp(0.050 x 0.75) = 98.63.

The hardest tier

A commodity trades at USD 75. The one-year forward is USD 84. Assume the net cost of carry (r + storage - convenience yield) stays constant across maturities. The two-year forward price is closest to:

  1. USD 93.00Why not: Adds the same dollar increment each year, but carry compounds proportionally.
  2. USD 94.08 - correct
  3. USD 105.37Why not: Compounds the one-year forward over two further years, double counting the first year.
  4. USD 66.96Why not: Discounts spot at the implied carry instead of growing it.

Net carry = ln(84/75) = 0.1133; F(2) = 75 x exp(2 x 0.1133) = 75 x 1.12² = 94.08.

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