How does the convenience yield affect commodity forward pricing, and what is the full formula?

Question

AcadiFi · Accepted Answer

The convenience yield is one of the trickiest concepts in commodity forwards because it represents a benefit that is invisible in the cash flows but very real in pricing.

## The Cost-of-Carry Model for Commodities

The general forward pricing formula for a commodity is:

**F_0 = S_0 x e^{(r + u - y) x T}**

Where:
- F_0 = forward price today
- S_0 = spot price today
- r = risk-free rate (continuously compounded)
- u = storage cost rate (as a continuous yield)
- y = convenience yield (continuous)
- T = time to maturity

**Why does each component work this way?**

- **r** (cost of financing): Holding a physical commodity ties up capital. The forward price must compensate the holder for this opportunity cost.
- **u** (storage costs): Physical commodities cost money to store — warehousing, insurance, spoilage. This pushes the forward price UP.
- **y** (convenience yield): Holding the physical commodity provides a non-monetary benefit — you can use it in production or sell it during a supply squeeze. This pulls the forward price DOWN.

## Example: Copper Forward

Atlas Metals is pricing a 6-month forward on copper.
- Spot price: $9,200 per metric ton
- Risk-free rate: 5.00% (continuous)
- Storage cost: 2.00% (continuous)
- Convenience yield: 3.50% (continuous)

F_0 = 9,200 x e^{(0.05 + 0.02 - 0.035) x 0.5}
F_0 = 9,200 x e^{0.035 x 0.5}
F_0 = 9,200 x e^{0.0175}
F_0 = 9,200 x 1.01765
F_0 = **$9,362.38**

Notice the net cost of carry is only 3.5% (= 5% + 2% - 3.5%), much less than the 7% it would be without the convenience yield.

## When Is the Convenience Yield High?

The convenience yield tends to spike when:
1. **Inventories are low** — holding physical stock is extremely valuable during shortages
2. **Supply disruptions** — geopolitical events, weather, logistics failures
3. **Seasonality** — natural gas before winter, agricultural commodities before harvest

When inventories are abundant, the convenience yield approaches zero and the market is in **contango** (F > S). When the convenience yield exceeds r + u, the market is in **backwardation** (F < S).

```mermaid
flowchart TD
    S["Spot Price S_0"] --> CC{"Cost of Carry"}
    CC -->|"+ Risk-free rate r"| UP["Pushes F UP"]
    CC -->|"+ Storage cost u"| UP
    CC -->|"- Convenience yield y"| DOWN["Pulls F DOWN"]
    UP --> F["Forward Price F_0"]
    DOWN --> F
    F --> CONTANGO["If r+u > y: Contango"]
    F --> BACKWARDATION["If y > r+u: Backwardation"]
```

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How does the convenience yield affect commodity forward pricing, and what is the full formula?

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