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Find a commodity's forward price when storing it costs money.

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

The idea

When storing a commodity costs money, the forward price must carry those costs, because an arbitrageur who buys and holds the commodity has to pay them. With storage as a continuous proportional cost λ a year, the forward price is F = S × e^((r + λ)T). Here S is the spot price, r the continuously compounded risk-free rate and T the time to expiry. With storage paid as fixed cash amounts, the forward price is F = S × e^(rT) + FV of storage. FV of storage is the sum of the payments, each compounded at the risk-free rate from its payment date to T. A payment made at the start is compounded for the full period, and one made at expiry is simply added. The usual errors are adding storage payments without compounding them, treating a payment as made at the wrong date, or subtracting storage costs. A convenience yield, if any, reduces the forward price.

Formulas

Commodity forward with storage paid in cash

F0 = S0 × e^(r × T) + FV of storage

S0
the spot price today
r
the continuously compounded risk-free rate, per year
T
the time to delivery, in years
FV of storage
each storage payment compounded at r from its payment date to T, summed

A payment at the start is compounded for the full period and one at expiry is simply added. With storage as a proportional rate λ, use S0 × e^((r + λ) × T) instead.

A practice question

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

A commodity has a continuous proportional storage cost lambda and no convenience yield. The no-arbitrage forward price is:

  1. F = S x exp((r - lambda) x T)Why not: Subtracts the storage cost; storage is an added cost of carry.
  2. F = S x exp((r + lambda) x T) - correct
  3. F = S x exp(r x T) - lambda x TWhy not: Treats lambda as a fixed amount subtracted at the end, not a proportional rate in the exponent.
  4. F = S x exp(lambda x T)Why not: Leaves out the financing cost of the cash used to buy the commodity.

Buying and storing for T costs interest r plus storage lambda, so the forward must equal S compounded at r + lambda.

A harder one

A commodity trades at USD 150. The continuously compounded rate is 10% and storage of USD 15 per unit is paid at the end of the year. The one-year forward price is closest to:

  1. USD 180.78 - correct
  2. USD 182.35Why not: Treats the storage payment as made up front and compounds it for the year.
  3. USD 179.35Why not: Discounts the end-of-year storage payment; it should be added at its payment date.
  4. USD 165.78Why not: Ignores the storage payment altogether.

F = 150 x exp(0.10) + 15 = 165.78 + 15 = 180.78. The payment at T needs no compounding.

The hardest tier

A commodity trades at USD 80 and the one-year forward is USD 88. The continuously compounded risk-free rate is 5% and there is no convenience yield. The storage cost rate consistent with no arbitrage is closest to:

  1. 5.00%Why not: Uses the simple percentage return rather than the continuously compounded one.
  2. 4.53% - correct
  3. 14.53%Why not: Adds the risk-free rate to the log return instead of subtracting it.
  4. 9.53%Why not: Takes the whole log return as storage cost, ignoring the interest cost.

ln(F/S) = ln(1.10) = 0.0953 = r + lambda, so lambda = 0.0953 - 0.05 = 4.53%.

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