Pricing Forwards, Futures, and Commodities

A trader sees gold spot at $2,400 and the six-month gold future at $2,460. Is that a 5% annualized arbitrage, a normal cost-of-carry market, or a signal about lease rates? The answer depends on whether the carry equation balances. Forward pricing on the FRM tests whether you can build that equation from spot, financing, income, and storage, and recognize when convenience yield breaks it.

A financial asset is a contractual or ownership claim, such as a bond, share or deposit. Holding a share gives exposure to a business and possible dividends; holding a bond gives a claim on its promised payments. A barrel of oil is a physical asset rather than a financial claim, even when an investor trades it for profit.

For forward pricing, a second classification matters. An investment asset is held by a substantial group primarily for investment, making spot holdings available for arbitrage. Securities and investment gold are examples; gold is an investment asset without being a financial claim.

Common mistakes

• Using the non-income formula on a dividend-paying asset. A stock index at 4,500 with a dividend yield of 2% and risk-free rate of 4% has a six-month forward of $4500 \times e^{0.01} = 4{,}545$, not $4500 \times e^{0.02} = 4{,}591$. Trap: 4,591 appears as the distractor for candidates who forget to net the yield.
• Confusing contango with a price forecast. A question states "the oil futures curve is in steep contango, what does the market expect?" The answer is "nothing about future prices; the curve reflects high financing and storage relative to convenience yield." Trap: "the market expects oil prices to rise" is wrong.
• Adding storage and lease rate. The lease rate already reflects net carry economics for a leasable commodity. Stacking u on top of -δ double-counts. Use $F_0 = S_0 e^{(r-\delta)T}$ when given the lease rate; use $F_0 = S_0 e^{(r+u-y)T}$ when given storage and convenience yield separately. Don't mix.

Bottom line

• No-arbitrage anchor: holding the asset must cost the same as locking in delivery via forward, so arbitrage sets the price.
• Non-income asset: $F_0 = S_0 e^{rT}$. The forward exceeds spot by exactly the financing cost of carrying the asset.
• Known cash income: $F_0 = (S_0 - I)e^{rT}$ where I is the present value of income to T. With continuous yield q: $F_0 = S_0 e^{(r-q)T}$.
• Cost of carry: $F_0 = S_0 e^{(r+u-y)T}$ (storage u, convenience yield y). Lease-rate form $F_0 = S_0 e^{(r-\delta)T}$ bundles both. Never stack one on the other.

Exam shortcut

When a forward problem lands, ask which carry inputs the stem gives. Risk-free rate alone → non-income form. Rate plus dividend or yield → $e^{(r-q)T}$. Rate plus storage plus convenience → cost-of-carry. Rate plus lease rate → lease-rate form. The trap is using the non-income formula by reflex when the asset pays income.

The full lesson (about 5,506 words, 37 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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