FRM Exam Part I · Commodity Forwards and Futures

Commodity Lease Rates and Forward Curves Explained

Updated 11 October 2026 · Fact-checked

The commodity lease rate is the return a lender of a commodity, such as gold, earns for lending it, paid in the commodity itself. It lowers the forward price: F = S × e^((r − δ)T). Given spot and forward prices you can solve for the implied lease rate, and the forward curve shows how F changes with maturity.

Understand Commodity Lease Rates and Forward Curves

Start with the no-arbitrage idea. If you buy a commodity today and sell it forward, you pay the spot price S, and you must fund that cash at the interest rate r. A forward price of S × e^(rT) would be fair if holding the commodity paid nothing. But many commodities can be lent out for a return.

Gold is the standard example. A holder such as a central bank can lend gold to a dealer. The dealer returns the gold later plus a fee. The fee is quoted as an annual rate and paid in gold. This is the lease rate, written δ (or δ_l). It acts like a dividend yield on the commodity. The lender gives up the use of the metal but earns the lease rate.

Because owning the physical commodity earns the lease rate, the cost of carry falls. The forward price is F = S × e^((r − δ)T), using continuous compounding. The higher the lease rate, the lower the forward price relative to spot. If δ > r, the forward price is below spot.

The forward curve (term structure) plots forward prices against maturity. If r > δ, the curve slopes up and is in contango. If δ > r, it slopes down and is in backwardation. For gold, the lease rate is usually small and below r, so the curve is usually upward sloping. The curve is read in reverse as well: from observed forward prices you extract the implied lease rate for each maturity, or between two maturities.

One caution. The lease rate is not observed directly in the same way a Treasury yield is. You usually infer it from the forward premium and the interest rate. For other commodities, the same formula holds, but the yield term is often called the convenience yield, and storage costs may also enter. Always follow the definition in the question.

Key formulas to remember

Forward price with lease rate (continuous compounding): F(0,T) = S(0) × e^((r − δ_l) × T); r is the continuously compounded risk-free rate and δ_l the continuously compounded lease rate, both annual. T is in years.
Implied lease rate from spot and forward: δ_l = r − (1 ÷ T) × ln(F ÷ S); The term (1 ÷ T) × ln(F ÷ S) is the annualised continuously compounded forward premium.
Annualised forward premium: (1 ÷ T) × ln(F ÷ S) = r − δ_l; Positive means contango. Negative means backwardation. The forward premium is the net cost of carry.
Forward ratio between two maturities: F(0,T2) ÷ F(0,T1) = e^((r − δ_l) × (T2 − T1)); Use it to find the implied lease rate between two forward dates. Rates are those for that period.
Discrete compounding version: F = S × (1 + r)^T ÷ (1 + δ_l)^T; Use only if the question states annual compounding. Do not mix with the continuous formula.

How to solve Commodity Lease Rates and Forward Curves questions

Use this method for any question on lease rates, forward prices or the shape of a commodity forward curve.

1Read the compounding convention. Default to continuous compounding unless the question says otherwise.
2Write down S, F, r, δ_l and T. Convert T to years, for example 6 months = 0.5.
3Decide what is unknown: forward price, lease rate, or the curve shape.
4For a forward price, compute (r − δ_l) × T, then apply e^( ) and multiply by S.
5For an implied lease rate, compute ln(F ÷ S), divide by T, and subtract the result from r.
6If two forwards are given, take the ratio of F(T2) to F(T1), use the time gap T2 − T1, and solve for the implied rate in that period.
7Judge the curve: r > δ_l means contango, r < δ_l means backwardation.
8Check that the answer is sensible: a higher lease rate must give a lower forward price, all else equal.

Quickest way: Forward premium shortcut

When to use it: Use for multiple-choice questions where the options are close. It avoids a full calculation.

For small rates, ln(F ÷ S) is close to (F − S) ÷ S. Use it to estimate the premium per year.
Lease rate ≈ r − annual premium. For example, a 2% premium with r = 5% gives about 3%.
Use your calculator's ln key for the final answer if the options differ by only a few basis points.
Sign check: if F > S and the lease rate is positive, r must exceed the lease rate. If F < S, the lease rate exceeds r.
Eliminate any option that moves the forward price the wrong way when the lease rate rises.

Common mistakes in Commodity Lease Rates and Forward Curves

Adding the lease rate to r instead of subtracting it.
Students link it to storage cost, which is added. The lease rate is a return earned, like a dividend.
Fix: Treat δ_l like a yield. The exponent is (r − δ_l), so a higher lease rate lowers the forward price.
Dropping the time factor when solving for the lease rate.
Students stop at ln(F ÷ S) and forget it covers T years, not one.
Fix: Divide ln(F ÷ S) by T before subtracting from r. This gives an annual rate.
Mixing simple and continuous compounding.
Some interest rates in the question are given as annual effective rates and others as continuous.
Fix: Convert all rates to the same convention first. Use e^(rT) with continuous rates and (1 + r)^T with annual rates.
Calling any upward sloping curve backwardation, or the reverse.
The names are easy to swap under time pressure.
Fix: Remember: contango means forward above spot and the curve rises. Backwardation means forward below spot and the curve falls.
Using spot-to-forward formula for the gap between two forwards.
Students use S and F(T2) with T2 when the question asks for the rate between T1 and T2.
Fix: Use the ratio F(T2) ÷ F(T1) and the time gap T2 − T1.
Assuming the lease rate is always positive and below r.
Gold is usually in contango, so students treat it as a rule.
Fix: The sign of r − δ_l decides the shape. Calculate it. Do not assume it.

Worked examples

Example 1

Gold spot is $1,900 per ounce. The 1-year forward price is $1,950. The continuously compounded risk-free rate is 4% per year. Find the implied annual lease rate.

Show the solution

Formula: δ_l = r − (1 ÷ T) × ln(F ÷ S).
F ÷ S = 1,950 ÷ 1,900 = 1.026316.
ln(1.026316) = 0.025976.
Divide by T = 1: annual forward premium = 0.025976, or 2.60%.
δ_l = 0.04 − 0.025976 = 0.014024.

Answer: The implied lease rate is about 1.40% per year.

Example 2

Gold spot is $2,000 per ounce, r = 5% and the lease rate is 1.5%, both continuously compounded. Find the 6-month and 12-month forward prices. Is the curve in contango or backwardation, and what is the implied rate between the two dates?

Show the solution

Net carry: r − δ_l = 0.05 − 0.015 = 0.035 per year.
6-month forward: F = 2,000 × e^(0.035 × 0.5) = 2,000 × e^0.0175.
e^0.0175 = 1.017654, so F(0.5) = $2,035.31.
12-month forward: F = 2,000 × e^0.035. e^0.035 = 1.035620, so F(1) = $2,071.24.
Shape: r > δ_l and the forward prices rise with maturity, so the curve is in contango.
Implied rate between dates: ln(2,071.24 ÷ 2,035.31) ÷ 0.5 = 0.0175 ÷ 0.5 = 0.035, so r − δ_l is 3.5%, the same as the flat input.

Answer: The 6-month forward is $2,035.31 and the 12-month forward is $2,071.24. The curve is in contango, with net carry of 3.5% per year.

Exam tips

Check the compounding convention before any calculation. Most FRM questions on this topic use continuous compounding.
If a question gives F and S and asks for the lease rate, the answer is always r minus the annualised forward premium. Do the ln step on your calculator.
When asked about curve shape, compare r and the lease rate. You do not need prices.
Watch for questions on who earns the lease rate. The lender of the physical commodity earns it, and it is paid in the commodity.
Expect a conceptual twist: higher lease rate means lower forward price and a flatter or downward sloping curve.

Practice questions from Commodity Forwards and Futures

Commodity Lease Rates and Forward Curves in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IICommodity Spot, Forward and Futures Markets

Commodity Lease Rates and Forward Curves: frequently asked questions

What is a commodity lease rate?

It is the annual return a lender of a commodity, such as gold, receives for lending it. It is paid in the commodity, not cash. In forward pricing it works like a dividend yield.

How do I derive the lease rate from gold forward prices?

Compute ln(F ÷ S) and divide by T to get the annual forward premium. Subtract that from the risk-free rate r. The result is the implied lease rate: δ_l = r − (1 ÷ T) × ln(F ÷ S).

Is the lease rate the same as the convenience yield?

They play the same role in the forward price formula, because both reduce the forward price. The lease rate is the market return from lending the commodity, mainly used for gold and silver. Convenience yield is the benefit of holding physical stock for other commodities.

Does a higher lease rate mean contango or backwardation?

A higher lease rate pushes the curve toward backwardation. If the lease rate is below r the curve is in contango. If it is above r, the forward price is below spot and the curve is in backwardation.