Commodity Forwards and Futures

Q: What are storage costs in the context of commodity futures?

Storage costs ($ u $) represent the physical expenses of holding a commodity, such as warehouse rent, insurance, and security.

Q: What is convenience yield and how does it affect commodity pricing?

Convenience yield ($ y $) is the non-monetary benefit of holding a physical commodity; it reduces the forward price.

Q: How do changes in storage costs and convenience yield impact the forward price?

Higher storage costs ($ u $) increase the forward price, whereas a higher convenience yield ($ y $) decreases the forward price.

Q: Why do investment commodities like gold usually have a convenience yield of zero?

Investment commodities are held for value storage rather than industrial consumption, so there is no operational benefit to physical possession.

Q: How is the forward price calculated if storage costs are a fixed dollar amount instead of a percentage?

$ F = (S + U) \times e^{rT} $, where $ U $ is the present value of the storage costs.

Q: What is Contango in commodity markets?

Contango is a market condition where the futures price is higher than the spot price ($ F > S $).

Q: What are the steps for Cash-and-Carry Arbitrage in commodities?

Borrow money, buy the physical commodity, short the futures contract, pay storage costs, and deliver the asset at maturity.

Q: How is the Commodity Lease Rate ($ \eta $) defined?

The lease rate is the return required by investors to lend the commodity, defined as $ \eta = r + u - y $.

FRM Part I · FRM · 6 min read

Welcome to Commodity Forwards and Futures!

Hello! If you’ve already studied stock or bond futures, you might think you’ve seen it all. But commodities like oil, gold, and corn are different beasts entirely. They take up space, they can spoil, and sometimes, just holding the physical item is more valuable than holding a piece of paper. In this chapter, we will explore how these physical realities change the way we price forward and futures contracts. Don't worry if this seems a bit "heavy" at first—we'll break it down piece by piece!

1. Commodities vs. Financial Assets: The Basics

Before we dive into the math, we need to understand why commodities aren't just "stocks that smell like oil." Financial assets (like stocks) provide dividends, but commodities have carrying costs.

Storage Costs: Unlike a digital share of Apple, 1,000 barrels of crude oil require a physical tank. You have to pay for that space, insurance, and security. This is a "leakage" or a cost of holding the asset.

Convenience Yield: This is a unique "benefit" of holding the physical commodity. Imagine you run a bakery and there is a sudden wheat shortage. If you own physical wheat, you keep baking. If you only own a futures contract for wheat delivered in three months, your ovens stay cold. That "peace of mind" and operational flexibility is the convenience yield.

Quick Review:
• Financial Assets: Often pay you to hold them (dividends/interest).
• Commodities: Usually cost you money to hold (storage) but offer a "hidden" benefit (convenience yield).

2. The Commodity Pricing Formula (Cost of Carry)

In the world of the FRM, the Cost of Carry model is your best friend. For a commodity, the forward price $ F $ is determined by the spot price $ S $, the risk-free rate $ r $, storage costs $ u $, and the convenience yield $ y $.

The general formula using continuous compounding is:
$ F = S \times e^{(r + u - y)T} $

Where:
$ S $: Current Spot Price
$ r $: Risk-free interest rate
$ u $: Storage costs (expressed as a percentage of the price per year)
$ y $: Convenience yield (expressed as a percentage per year)
$ T $: Time to maturity

Analogy Time: Think of the formula like a backpack. The spot price is the weight you start with. Interest ($ r $) and storage ($ u $) are extra items you have to carry (they increase the future price). The convenience yield ($ y $) is like a helium balloon tied to the bag that makes it feel lighter (it decreases the future price).

Key Takeaway: If storage costs go up, the forward price goes up. If the convenience yield goes up, the forward price goes down.

3. Investment vs. Consumption Commodities

The FRM curriculum distinguishes between two types of commodities because their pricing behavior differs:

Investment Commodities

These are held primarily for investment purposes (like Gold and Silver). Because people don't "consume" them in the same way as oil, the convenience yield is usually zero.
Formula: $ F = S \times e^{(r + u)T} $

Consumption Commodities

These are used in production (like Oil, Copper, or Corn). These always have a convenience yield because manufacturers need to keep inventories to avoid production shutdowns. For these, we use the full formula including $ y $.

Common Mistake to Avoid: Don't assume storage costs are always a percentage. If storage is a fixed dollar amount $ U $, the formula changes to: $ F = (S + U) \times e^{rT} $. Always read the question carefully to see if storage is in \$ or %!

4. Backwardation and Contango

These two terms are "exam favorites." They describe the shape of the forward curve (the relationship between the spot price and the future price).

Contango:
This happens when the futures price is higher than the spot price ($ F > S $). This is the "normal" state for most commodities because storage and interest costs make it more expensive to get the item later than to have it now.
Memory Aid: Contango = Costly to carry.

Backwardation:
This happens when the futures price is lower than the spot price ($ F < S $). This occurs when the convenience yield ($ y $) is very high—usually because there is a supply shortage. People want the physical goods right now and are willing to pay a premium for immediate delivery.
Memory Aid: Backwardation = Better to have it now.

Did you know? In 2020, oil prices briefly went into "Super Contango" because there was so much oil and nowhere to store it. The cost of storage became so high that the spot price crashed, while future prices remained much higher.

5. Arbitrage in Commodities

If the market price of a futures contract doesn't match our "Cost of Carry" formula, an arbitrage opportunity exists. There are two main strategies:

Cash-and-Carry Arbitrage

When to use: When the futures price is too high ($ F > S \times e^{(r+u)T} $).
Step-by-Step:
1. Borrow money at rate $ r $.
2. Buy the physical commodity at price $ S $.
3. Short (sell) a futures contract at price $ F $.
4. Pay the storage costs $ u $.
5. At maturity, deliver the commodity to fulfill the futures contract. The difference is your risk-free profit!

Reverse Cash-and-Carry Arbitrage

When to use: When the futures price is too low ($ F < S \times e^{(r+u)T} $).
Note: This only works for investment commodities like gold. You can't easily do this with oil because you can't "short sell" physical oil easily if you don't already own it.

Key Takeaway: Arbitrage acts like a magnet, pulling the market price back toward the theoretical price calculated by our formula.

6. The Commodity Lease Rate

Sometimes, instead of talking about convenience yields, the curriculum mentions the Lease Rate.
The lease rate ($ \eta $) is the return required by an investor to lend out their commodity. If you own gold and "lease" it to someone else, they pay you interest in the form of more gold.

The relationship is: $ Lease Rate (\eta) = r + u - y $.
This allows us to write the forward price as: $ F = S \times e^{(r - \eta)T} $.

Quick Summary Table:
• Interest ($ r $): Increases Forward Price
• Storage ($ u $): Increases Forward Price
• Convenience Yield ($ y $): Decreases Forward Price
• Lease Rate ($ \eta $): Decreases Forward Price

Final Encouragement

You’ve just covered the core mechanics of commodity pricing! The math looks intimidating with the $ e $ and the exponents, but remember: it's all just accounting for the costs and benefits of holding "stuff" over time. Focus on the relationship between $ r, u, $ and $ y $, and you'll be well-prepared for any commodity question the FRM exam throws at you. Keep going, you're doing great!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are storage costs in the context of commodity futures?

Storage costs ($ u $) represent the physical expenses of holding a commodity, such as warehouse rent, insurance, and security.

What is convenience yield and how does it affect commodity pricing?

Convenience yield ($ y $) is the non-monetary benefit of holding a physical commodity; it reduces the forward price.

What is the general Cost of Carry formula for a commodity with continuous storage costs and convenience yield?

$ F = S \times e^{(r + u - y)T} $

How do changes in storage costs and convenience yield impact the forward price?

Higher storage costs ($ u $) increase the forward price, whereas a higher convenience yield ($ y $) decreases the forward price.

Why do investment commodities like gold usually have a convenience yield of zero?

Investment commodities are held for value storage rather than industrial consumption, so there is no operational benefit to physical possession.

How is the forward price calculated if storage costs are a fixed dollar amount instead of a percentage?

$ F = (S + U) \times e^{rT} $, where $ U $ is the present value of the storage costs.

What is Contango in commodity markets?

Contango is a market condition where the futures price is higher than the spot price ($ F > S $).

What is Backwardation and what causes it?

Backwardation is when the futures price is lower than the spot price ($ F < S $), usually caused by a high convenience yield during supply shortages.

What are the steps for Cash-and-Carry Arbitrage in commodities?

Borrow money, buy the physical commodity, short the futures contract, pay storage costs, and deliver the asset at maturity.

How is the Commodity Lease Rate ($ \eta $) defined?

The lease rate is the return required by investors to lend the commodity, defined as $ \eta = r + u - y $.

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