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Work through the cost of carry model to see how commodity forward prices and no-arbitrage bounds respond to convenience yields and to storage costs.

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

The idea

In the cost of carry model, the forward price of a commodity held for storage and use is F = S × e^((r + λ − c)T). Here λ is the storage cost rate and c the convenience yield. A higher storage cost raises the forward price, and a higher convenience yield lowers it, so the curve shifts toward backwardation. Cash-and-carry arbitrage gives an upper bound for a commodity forward, which is Fmax = S × e^((r + λ)T). If the forward is above that bound, arbitrage profits exist. There is no equally firm lower bound, because owners of the physical commodity will not sell it just to capture a gap, which is why the convenience yield appears. Turning the model round gives the implied convenience yield, which can be read from a quoted forward as c = r + λ − ln(F ÷ S) ÷ T. It should not be negative. The usual error is applying a two-sided bound to a commodity that is consumed.

Formulas

Implied convenience yield

c = r + λ − ln(F0 ÷ S0) ÷ T

r
the continuously compounded risk-free rate, per year
λ
the storage cost, as a continuous rate per year
F0
the market forward price
S0
the spot price today
T
the time to delivery, in years

It should not be negative: a forward above full carry is an arbitrage, not a negative yield. A forward below carry is read as a convenience yield, not as an arbitrage, for a consumed commodity.

Upper no-arbitrage bound on a commodity forward

upper bound = S0 × e^((r + λ) × 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
T
the time to delivery, in years

F0 cannot exceed this, because cash-and-carry would profit. There is no matching lower bound when holders will not lend or sell the physical commodity.

A practice question

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

In the cost-of-carry model, a higher convenience yield, with spot, interest rate and storage costs unchanged, makes the forward curve:

  1. higher and more likely to slope upward (contango).Why not: Gets the sign wrong: convenience yield is subtracted in the formula.
  2. lower and more likely to slope downward (backwardation). - correct
  3. unchanged, since convenience yield affects only spot.Why not: Convenience yield is a direct input to the forward formula.
  4. higher, because it compensates holders for risk.Why not: Mixes the convenience yield with a risk premium; it is a benefit of holding the commodity.

F = S x exp((r + lambda - c) x T): c lowers the net carry, so the forward is lower and the curve flatter or downward-sloping.

A harder one

A commodity trades at USD 100 with r = 5% and storage costs of 3% (continuous, per year). The cost-of-carry upper bound for the one-year forward is 108.33. Which observed one-year forward price signals a riskless arbitrage?

  1. USD 108Why not: Inside the bound (below 108.33); no cash-and-carry profit, and the gap implies a small convenience yield.
  2. USD 104Why not: Below the bound: a convenience yield of about 4% is consistent with it, so it offers no arbitrage.
  3. USD 112 - correct
  4. USD 98Why not: Below spot, but a high convenience yield can justify that; only the upper bound is firm for a consumed commodity.

Cash-and-carry gives F <= 108.33. Only 112 exceeds it, so selling the forward and buying and storing the commodity locks in a profit. Lower values imply positive convenience yields, not arbitrage.

The hardest tier

A commodity trades at USD 70, the risk-free rate is 4%, and T is one year. Storage costs rise from 2% to 3% and the convenience yield rises from 1% to 4%, both per year and continuous. The new forward price is closest to:

  1. USD 72.13 - correct
  2. USD 74.33Why not: Applies the higher storage cost but not the higher convenience yield.
  3. USD 71.41Why not: Applies the higher convenience yield but not the higher storage cost.
  4. USD 78.14Why not: Adds the convenience yield instead of subtracting it.

Before: net carry = 0.04 + 0.02 - 0.01 = 0.05, F = 73.59. After: 0.04 + 0.03 - 0.04 = 0.03, F = 70 x exp(0.03) = 72.13. The yield rise outweighs the storage rise, so F falls.

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