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Show how the lease rate sets the no-arbitrage values of futures and forwards on commodities, and define the lease rate.

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

The idea

The lease rate is the return a lender of a commodity receives, paid in units of the commodity. It plays the role a dividend yield plays for a stock. It exists mainly for gold and silver, which central banks and others hold and will lend. With lease rate l, the no-arbitrage forward for a commodity that can be lent out is F = S × e^((r − l)T). Turned around, the same relation lets you back out the lease rate from market prices as l = r − ln(F ÷ S) ÷ T. A higher lease rate lowers the forward price. If F is above this value, buy the commodity, lend it out and sell forward. If F is below, borrow it, sell it, invest the cash and buy forward. The usual error is to use the lease rate where a convenience yield belongs, or to forget to annualise ln(F ÷ S).

Formulas

Lease rate implied by a commodity forward

l = r − ln(F0 ÷ S0) ÷ T

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

Rearranged from F0 = S0 × e^((r − l) × T). Annualise ln(F0 ÷ S0) by dividing by T; a forward below spot implies a lease rate above r.

A practice question

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

The lease rate on a commodity is best defined as:

  1. the cost of storing the commodity in a warehouse.Why not: This is a storage cost, which raises the forward price; the lease rate lowers it.
  2. the return a lender of the commodity receives, paid in units of the commodity, analogous to a dividend yield. - correct
  3. the interest the borrower pays on the cash proceeds from selling the commodity.Why not: Confuses the lease payment, made in commodity units, with interest on cash.
  4. the difference between the futures price and the spot price.Why not: Describes the basis; the lease rate is a rate of return, not a price gap.

Gold is lent by central banks and others for a fee paid in gold. The lease rate behaves like a dividend yield in the forward formula F = S x exp((r - l) x T).

A harder one

Gold is USD 1,900, the risk-free rate is 5% and the lease rate is 2%. The one-year forward is USD 1,970. Which trade captures a riskless profit?

  1. Borrow cash to buy gold, lend the gold to earn the lease rate, and sell gold forward. - correct
  2. Borrow gold, sell it spot, invest the proceeds and buy gold forward.Why not: This is the reverse trade, which works when the forward is below S x exp((r - l) x T).
  3. Buy gold forward and sell gold spot, holding the proceeds in cash.Why not: Uses an unhedged forward purchase and ignores that gold can be lent for income.
  4. No trade; the forward is within the no-arbitrage bounds.Why not: The no-arbitrage value is 1,900 x exp(0.03) = 1,957.8, below 1,970.

No-arbitrage forward = 1,900 x exp(0.03) = 1957.9. The actual 1,970 is too high, so sell it forward and own the gold (financed, and lent out for the lease fee).

The hardest tier

Silver is USD 30.0 per ounce and the one-year forward is USD 29.4. The continuously compounded risk-free rate is 3% per year. The implied lease rate is closest to:

  1. 0.98%Why not: Adds the (negative) log return, changing the sign of its effect.
  2. 5.02% - correct
  3. -2.02%Why not: Reports the log return itself as the lease rate, ignoring the risk-free rate.
  4. -5.02%Why not: Uses ln(F/S) - r, the negative of the correct formula.

l = r - ln(F/S)/T = 0.03 - (-0.0202) = 5.02%. A forward below spot means the lease rate exceeds the risk-free rate.

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