## **Learning Outcomes**

This article explains how commodity forward prices differ from financial forward prices because commodities can be costly to store, hard to transport, and valuable to hold as inventory. After reading, you will be able to:

- Price a commodity forward using the cost-of-carry model with storage costs and convenience yield
- Use cash-and-carry and reverse cash-and-carry arguments to identify commodity forward arbitrage and bounds
- Interpret the roles of storage, convenience yield, and lease rates in contango and backwardation

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand commodity forwards and futures pricing with a focus on the following syllabus points:

- Define and apply storage costs, carry markets, lease rate, and convenience yield
- Explain and apply the formula for pricing commodity forwards (including storage costs and convenience yields)
- Describe arbitrage transactions in commodity forwards and determine no-arbitrage values and no-arbitrage bounds

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. A storable commodity has spot price \$50, present value of storage costs \$2 over the next year, risk-free rate 4% per year, and no convenience yield. The no-arbitrage 1-year forward price is closest to:
   a) \$50.00  
   b) \$52.00  
   c) \$54.08  
   d) \$54.16

2. Calculate the three-month forward price of a commodity if $S_0 = \$3.00$, the effective monthly interest rate is 1.5%, and the present value of storage costs over three months is \$0.09.

3. In one or two sentences, explain why convenience yield can cause commodity forward prices to be lower than “spot plus interest plus storage.”

4. True or False (brief explanation required): The upper bound on a commodity forward price (with storage costs and convenience yield) depends on the size of the convenience yield.

## **What Do You Know Already?**

- Can you write the cost-of-carry forward pricing equation for a commodity with storage costs (using the present value of storage)?
- Do you remember when cash-and-carry vs reverse cash-and-carry arbitrage applies in commodity markets?
- Can you explain, in plain language, what convenience yield represents and who can earn it?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Commodity forwards and futures are core risk management tools for firms exposed to input costs (e.g., airlines buying jet fuel, food producers buying grains, manufacturers buying industrial metals). For these firms, the forward curve is not “just interest rates”: it embeds storage economics, inventory scarcity, and the value of having the physical commodity available when needed.

For FRM Part 1, the key technical idea is that commodities often behave differently from financial assets in no-arbitrage pricing. Storage costs push forward prices up. Convenience yield (a non-cash benefit of inventory) pushes forward prices down. Because convenience yield is not directly observable and is not earned by all market participants, commodity forwards often have **no-arbitrage bounds** rather than a single strict no-arbitrage price.

> **Key Term: Cost of carry**  
> The net cost of holding a physical commodity to maturity, typically financing (interest) plus storage costs, offset by benefits such as convenience yield or lease income.

## **Commodities versus financial assets**

Commodity forwards are priced with the same broad no-arbitrage logic as financial forwards, but several commodity features weaken (or reshape) the tight spot-forward link you may be used to from equities or bonds:

1. **Storage costs can be large and operationally complex**  
   Commodities may require warehousing, special handling, insurance, and may spoil or degrade. Financial assets have negligible storage cost.

2. **Location and transport matter**  
   Commodity prices can vary by delivery location because transport can be expensive (notably for natural gas and electricity). Financial assets are transferred electronically at minimal cost.

3. **Shorting a commodity can require borrowing the physical good**  
   For some commodities (especially “investment commodities” like gold), borrowing/lending markets exist and borrowing costs are expressed via a **lease rate**.

4. **Sources of return differ**  
   Financial assets may provide income (dividends, coupons) and risk-related expected returns. Many commodities provide only price changes and are often described as **mean-reverting** over long horizons due to supply responses and substitution effects.

> **Key Term: Carry market**  
> A market for a storable commodity where it is feasible to hold inventory over time; forward prices therefore reflect carrying costs such as storage and financing.
>
> **Key Term: Storage costs**  
> The costs of physically holding a commodity (warehousing, insurance, spoilage/care). In forward pricing problems, these are often treated as known cash costs over the contract’s life.

### Storable versus non-storable matters

- **Storable commodities** (corn, oil, many metals) support classic cost-of-carry arbitrage arguments because you can buy, store, and deliver later.
- **Non-storable (or very hard-to-store) commodities** (electricity, some weather exposures) do not support the same arbitrage mechanism; pricing becomes more about supply-demand expectations and risk premia than about replication.

## **Cost of carry pricing with storage costs**

For a storable commodity with known storage costs, the forward price must compensate the party effectively holding the commodity until delivery. In the simplest FRM setup:

- $S_0$ = current spot price per unit
- $U$ = **present value** (PV) of storage costs over $[0,T]$ per unit
- $r$ = risk-free rate (effective annual, consistent with $T$)
- $F_{0,T}$ = forward price for delivery at $T$

The cost-of-carry forward price with storage costs (and no convenience yield) is:

$$F_{0,T} = (S_0 + U)(1+r)^T$$

Intuition:

- The long-forward investor could instead buy the commodity today, pay storage, and finance the total cost until $T$.
- The short-forward party, if replicating by buying and carrying inventory, needs to recover both financing and storage.

### Timing of storage costs in exam questions

In many FRM questions, storage costs are given as a schedule (monthly, quarterly) but you are also given (or can compute) their **present value** $U$. Once you have $U$, you add it to $S_0$ and then compound at the risk-free rate to maturity.

> **Key Term: Forward price vs value of a forward**  
> The **forward price** is the delivery price that makes a new forward contract worth zero today. The **value** of an existing forward can be positive or negative as market forward prices move.

For reference, if you entered a forward at delivery price $K$, then (with discrete compounding) the time-0 value to the long is:

$$V_0 = \frac{F_{0,T} - K}{(1+r)^T}$$

A newly initiated forward has $K = F_{0,T}$, hence $V_0 = 0$.

### Worked Example 1.1

Current spot price for corn is \$3/bushel. The effective monthly interest rate is 1.5%, and the present value of storage costs over the next three months is \$0.09/bushel. Calculate the three-month forward price.

> **Answer:**
>
> 1. Identify inputs (per bushel):
>    - $S_0 = 3.00$
>    - $U = 0.09$ (already PV over the three months)
>    - Monthly effective rate = 0.015
>    - $T = 3$ months
> 2. Use the storage-cost forward pricing equation with period-by-period compounding:  
>    $$F_{0,T} = (S_0 + U)(1+r)^T = (3.00 + 0.09)(1.015)^3$$
> 3. Compute:
>    - $3.00 + 0.09 = 3.09$
>    - $(1.015)^3 \approx 1.045678$
>    - $F_{0,T} \approx 3.09 \times 1.045678 \approx 3.231$
>
> Final answer: the forward price is approximately **\$3.23 per bushel**.

## **Arbitrage trades and no-arbitrage forward prices**

Forward pricing is enforced (when feasible) by arbitrage. For storable commodities, the standard mechanism is **cash-and-carry arbitrage**.

> **Key Term: Cash-and-carry arbitrage**  
> An arbitrage where you buy the commodity in the spot market (funded by borrowing), carry it to maturity (paying storage), and sell it forward. It is used when the market forward price is “too high.”

### Cash-and-carry when the forward is overpriced

If the market forward price $F^{mkt}$ is greater than the theoretical no-arbitrage price:

$$F^{mkt} > (S_0+U)(1+r)^T$$

an arbitrageur can:

1. Borrow $(S_0 + U)$ at rate $r$ for $T$
2. Buy the commodity spot (pay $S_0$)
3. Pay storage costs as incurred (total PV is $U$)
4. Short the forward at $F^{mkt}$
5. At $T$, deliver the commodity into the forward, receive $F^{mkt}$, repay the loan $(S_0+U)(1+r)^T$

Profit per unit at maturity:

$$\Pi_T = F^{mkt} - (S_0+U)(1+r)^T$$

### Reverse cash-and-carry when the forward is underpriced

If the market forward price is too low relative to the replication:

$$F^{mkt} < (S_0+U)(1+r)^T$$

then a trader would like to:

1. Short (sell) the commodity spot today
2. Invest the proceeds at the risk-free rate
3. Go long the forward to buy back at $T$

In practice, step 1 can be difficult for many commodities because it may require borrowing the physical commodity. This is one reason commodity forward pricing often becomes a **range** once convenience yield is introduced.

### Worked Example 2.1

A storable commodity has $S_0 = \$100$ per unit. The present value of storage costs over six months is $U = \$1.50$ per unit. The six-month risk-free rate is 2% (effective for the six-month period). The market six-month forward price is \$105.00. Determine whether an arbitrage exists and compute the arbitrage profit per unit (ignoring transaction costs).

> **Answer:**
>
> 1. Compute the no-arbitrage forward price:  
>    $$F^* = (S_0 + U)(1+r) = (100 + 1.50)(1.02) = 101.50 \times 1.02 = 103.53$$
> 2. Compare to the market forward:
>    - $F^{mkt} = 105.00$
>    - Since $105.00 > 103.53$, the forward is overpriced.
> 3. Execute cash-and-carry: borrow \$101.50, buy the commodity, store it, short the forward.
> 4. Profit per unit at maturity:  
>    $$\Pi = F^{mkt} - F^* = 105.00 - 103.53 = 1.47$$
>
> Final answer: **Yes**, an arbitrage exists; profit is approximately **\$1.47 per unit** at maturity (before costs and constraints).

## **Convenience yield and forward price bounds**

Storage costs are cash expenses. Convenience yield is different: it represents a benefit of **having inventory available**.

Examples of convenience yield in practice:

- A refinery holding crude oil avoids costly shutdowns if supply is disrupted.
- A manufacturer holding copper can keep production running even when shipments are delayed.
- A food processor holding grains can meet delivery commitments during temporary shortages.

> **Key Term: Convenience yield**  
> The nonmonetary benefit from holding physical inventory (often related to avoiding stockouts or production disruptions). It is typically available to producers and users of the commodity, not to a passive investor.

### Forward pricing with storage and convenience yield

Using the notation from the syllabus reading:

- $Y$ = annualized convenience yield (as a rate, proportional to commodity value)

A commonly used FRM form is:

$$F_{0,T} = (S_0 + U)\left(\frac{1+r}{1+Y}\right)^T$$

Convenience yield acts like a “benefit of carry,” reducing the forward price relative to the pure cost-of-carry level.

### Why we often get bounds, not a single price

A passive investor typically cannot capture convenience yield, because they do not run a production process that benefits from immediate access to the commodity. As a result:

- The **upper bound** comes from a long who can always buy spot, store, and finance instead of paying an overly high forward.
- The **lower bound** comes from a short (an inventory holder) who would refuse to sell forward too cheaply because selling forward reduces (or gives up) their convenience yield.

The FRM arbitrage-free bounds (with storage and convenience yield) are:

$$(S_0 + U)\left(\frac{1+r}{1+Y}\right)^T \leq F_{0,T} \leq (S_0 + U)(1+r)^T$$

Key implications:

- **Upper bound does not include $Y$** (the buyer does not receive convenience yield from holding a forward).
- **Lower bound declines as $Y$ rises** (inventory is more valuable to hold, so the short needs more compensation to give it up).

### Worked Example 3.1

A commodity has spot price \$80 per unit. The present value of storage costs over one year is \$2 per unit. The 1-year risk-free rate is 5% (effective annual). The observed 1-year forward price is \$79.00.

1. Compute the implied annual convenience yield $Y$.
2. Compute the arbitrage-free bounds on the forward price using this implied $Y$.

> **Answer:**  
> Given: $S*0 = 80$, $U = 2$, $r = 0.05$, $T = 1$, $F*{0,1} = 79$
>
> **1) Implied convenience yield**
>
> Start from:  
> $$F_{0,T} = (S_0+U)\left(\frac{1+r}{1+Y}\right)^T$$
>
> With $T = 1$:  
> $$79 = 82\left(\frac{1.05}{1+Y}\right)$$
>
> Solve:  
> $$\frac{79}{82} = \frac{1.05}{1+Y}$$  
> $$1+Y = 1.05\left(\frac{82}{79}\right)$$  
> $$1+Y \approx 1.05 \times 1.037975 = 1.089874$$  
> $$Y \approx 0.089874 \approx 8.99\%$$
>
> **2) Arbitrage-free bounds**
>
> Upper bound:  
> $$F_{max} = (S_0+U)(1+r)^T = 82 \times 1.05 = 86.10$$
>
> Lower bound (using implied $Y$):  
> $$F_{min} = (S_0+U)\left(\frac{1+r}{1+Y}\right)^T = 82 \times \frac{1.05}{1.089874} \approx 79.00$$
>
> Final answers:
>
> - Implied convenience yield is approximately **8.99%**
> - Bounds are approximately **\$79.00 ≤ F ≤ \$86.10**

### **Exam Warning**

> The most common exam errors are:
>
> 1. **Mixing PV and FV of storage costs**: if the problem gives PV of storage costs ($U$), add it to $S_0$ and then compound. If it gives future storage amounts, discount them first to get $U$.
> 2. **Putting convenience yield in the upper bound**: the upper bound is $(S_0+U)(1+r)^T$ and does **not** include $Y$.

## **Lease rates and borrowing commodities**

Some commodities (notably precious metals) are sometimes held for investment as well as consumption. When an active borrowing/lending market exists for the physical commodity, a party can borrow the commodity to short it, and the borrowing cost is described by a **lease rate**.

> **Key Term: Lease rate**  
> The rate (in commodity lending markets) paid by the borrower of the physical commodity; it is analogous to an income yield in forward pricing because it is a benefit to holding the commodity and lending it out.

If $\delta$ is the lease rate, the forward price for an “investment commodity” with an active lease market is:

$$F_{0,T} = S_0\left(\frac{1+r}{1+\delta}\right)^T$$

Interpretation:

- Higher $\delta$ makes holding the commodity more attractive (because you can lend it), so the forward price is lower.
- This is mathematically similar to pricing equity forwards with a dividend yield.

Lease rates can also change with supply available for lending and with demand to borrow, so they can vary significantly over time.

## **Forward price, expected future spot, and market states**

### Forward price is not always the expected future spot price

A forward price is a **risk-neutral / no-arbitrage** object (built from financing, storage, and carry benefits). The expected future spot price $E(S_T)$ is a **statistical expectation** under the real-world probability measure.

One syllabus result links them using a risk-adjusted discount rate $\alpha$ (for the risky $S_T$ cash flow):

$$\frac{F_{0,T}}{(1+r)^T} = \frac{E(S_T)}{(1+\alpha)^T}$$

Rearranging:

$$F_{0,T} = E(S_T)\left(\frac{1+r}{1+\alpha}\right)^T$$

If $\alpha > r$ (a positive risk premium is required to bear commodity price risk), then $F_{0,T} < E(S_T)$. If $\alpha < r$, then $F_{0,T} > E(S_T)$.

### Normal backwardation and contango

These terms are used in two related ways:

- Relative to **expected future spot**:
  - **Normal backwardation**: futures/forward price is below $E(S_T)$
  - **Contango**: futures/forward price is above $E(S_T)$

- Relative to **current spot** (common market usage):
  - Backwardation: futures/forward price below $S_0$
  - Contango: futures/forward price above $S_0$

> **Key Term: Normal backwardation and contango**  
> Normal backwardation means futures are below the expected future spot price; contango means futures are above it. In practice, the terms are also often applied relative to today’s spot price.

### How storage and convenience yield shape the curve

- High storage costs (large $U$) tend to push forward prices up, supporting contango.
- High convenience yield (large $Y$) tends to push forward prices down, supporting backwardation.
- Inventory levels often connect the two: low inventories are commonly associated with high convenience yield because the value of “having the commodity on hand” rises when scarcity risk rises.

From a risk management viewpoint, this matters because firms hedging through rolling futures positions may experience gains or losses from the term structure (often discussed as roll return), even if spot prices are unchanged.

## **Summary**

Commodity forward pricing starts from the same no-arbitrage principle used for financial forwards, but commodities introduce economically large frictions and benefits. Storage costs and financing costs create a cost of carry that tends to raise forward prices above spot. For storable commodities, this supports cash-and-carry arbitrage when forwards are “too high.”

Convenience yield is the main commodity-specific adjustment. It reflects the value of holding physical inventory to support operations and manage short-term scarcity. Because convenience yield is not a cash flow available to all traders, it often leads to **no-arbitrage bounds** rather than a single tight price relationship. High convenience yield can rationalize forward prices that appear low relative to spot-plus-carry and is closely linked to backwardation.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Commodities differ from financial assets due to storage, transport, shorting/borrowing frictions, and sources of return
- Storage costs enter commodity forward pricing through the cost-of-carry model: $F_{0,T} = (S_0+U)(1+r)^T$
- Cash-and-carry arbitrage applies when the forward is above the cost-of-carry level; reverse cash-and-carry applies when it is below (subject to shorting constraints)
- Convenience yield is a nonmonetary inventory benefit that reduces forward prices: $F_{0,T} = (S_0+U)\left(\frac{1+r}{1+Y}\right)^T$
- With convenience yield, arbitrage-free prices lie in a range:  
  $$(S_0 + U)\left(\frac{1+r}{1+Y}\right)^T \leq F_{0,T} \leq (S_0 + U)(1+r)^T$$
- Lease rates can affect forward prices for investment commodities: $F_{0,T} = S_0\left(\frac{1+r}{1+\delta}\right)^T$
- Forward prices can differ from expected future spot prices due to risk premia; contango and backwardation can be defined relative to spot or expected spot

## **Key Terms and Concepts**

- Cost of carry
- Carry market
- Storage costs
- Forward price vs value of a forward
- Cash-and-carry arbitrage
- Reverse cash-and-carry arbitrage
- Convenience yield
- Arbitrage-free bounds
- Lease rate
- Normal backwardation
- Contango


Currencies and commodities - Commodity forwards, storage, and convenience yield