Cost-of-Carry Pricing: Why the Forward Price Is What It Is

We’ve said three times now that the forward price $K$ usually isn’t equal to today’s spot price — and that it’s set to make the contract fair. Time to make good on that promise. There’s a single, beautiful argument that nails the fair forward price exactly, with no guessing about where prices are headed: the cost of carry. The punchline is that a forward price is just today’s spot price plus the cost of carrying the asset to delivery — and if it’s anything else, a risk-free money machine appears and traders smash it back into line. This lesson is that argument, end to end.

Before you read — take a guess

Pretest your instincts. Gold spot is $2,000. Storing and insuring it for a year is free, and the risk-free interest rate is 5%. What SHOULD the one-year forward price of gold be, roughly?

Exactly $2,000 — the forward price always equals the spot priceBelow $2,000, because future money is worth lessAbout $2,100 — spot grown by the 5% you forgo by tying up cash for a yearWhatever the market guesses gold will be worth in a year

The core idea: two ways to own gold in a year

Analogy. Suppose you definitely want a gold bar in your safe one year from now. There are two routes to get there:

Route A — buy forward. Enter a forward today to buy the bar in a year at price $K$ . Pay nothing now; pay $K$ at delivery.
Route B — buy-and-carry. Buy the bar today at spot $S_0$ , borrowing the cash, and just hold it for the year. At year-end you own the bar and owe the loan back with interest.

Both routes leave you holding one identical gold bar in one year. If two strategies deliver the exact same thing, they must cost the same today — otherwise you could do the cheap one and undo the expensive one for free money. That equality is the forward pricing formula. Let’s grind it out.

Deriving the formula by replication

Definition. The cost of carry is everything it costs (net) to hold the underlying from now until delivery: the financing cost (interest on the money tied up) plus storage and insurance, minus any income the asset throws off while you hold it (dividends, interest) and minus any convenience yield (the intangible benefit of having the physical asset on hand). For a simple non-income asset with rate $r$ over time $T$ and continuous compounding:

F_0 = S_0 \, e^{rT}.

For the general case, gather the carry costs $c$ (financing + storage) and the yields $y$ (income + convenience), as annual rates:

F_0 = S_0 \, e^{(c - y)\,T}.

In words you’ll never forget: forward = spot + cost of carrying − benefit of holding. When carry costs dominate, the forward sits above spot (the asset is in contango); when yields dominate, the forward sits below spot (backwardation) — the very shapes we chart next lesson.

The slider below builds exactly this. Start from spot, stack on financing and storage, subtract convenience yield, and watch the fair forward price assemble — and flip below spot when the convenience yield gets big enough:

Forward = spot + carry − convenience yield$106.00

Spot price: $100.00
+ Financing (interest): $5.00
+ Storage + insurance: $2.00
− Convenience yield: $1.00
= Fair forward price: $106.00

Financing rate (per year)5.0%

Storage cost (per year)2.0%

Convenience yield (per year)1.0%

Forward = spot + financing + storage − convenience yield, all over the time to delivery. Push convenience yield high enough and the forward drops below spot — that is exactly when an upward (contango) curve flips to a downward (backwardation) one.

Discrete vs continuous, don't panic

Textbooks write $F_0 = S_0 e^{rT}$ (continuous compounding) or $F_0 = S_0(1+r)^T$ (annual compounding). They’re two dialects of the same sentence — money grows at the financing rate over the holding period. We’ll use whichever keeps the arithmetic clean; the logic (spot grown by net carry) never changes.

Worked example — a non-income financial asset

A non-dividend stock trades at a spot of $100, so $S_0 = 100$ . The risk-free rate is 6% a year (use simple annual compounding for clarity), and a forward expires in one year. Storage and insurance are zero (it’s just a share).

F_0 = S_0(1 + r) = 100 \times 1.06 = 106.

The fair one-year forward price is $106. Why exactly $106 and not, say, $110? Because at $110 a free lunch appears — which we’ll now expose.

The enforcement mechanism: cash-and-carry arbitrage

The formula isn’t a polite suggestion; it’s enforced by ruthless arbitrageurs. Two trades do the policing.

When the forward is too EXPENSIVE → cash-and-carry

Say the forward is quoted at $110, above the fair $106. An arbitrageur pounces:

Borrow $100 at 6% and buy the stock today at spot.
Go short the forward at $110, locking in a sale price.
At expiry: deliver the stock for $110, repay the loan of $106 ($100 + $6 interest).
Pocket $4, risk-free, with no money of their own down.

Step	Cash flow now	Cash flow at expiry
Borrow $100	+$100	−$106 (repay with interest)
Buy stock at spot	−$100	(hold it)
Short forward at $110	$0	+$110 (deliver stock)
Net	$0	+$4 risk-free

Every arbitrageur does this, selling forwards en masse — which pushes the forward price down until the $4 profit vanishes at $106. That selling pressure is the formula being enforced.

When the forward is too CHEAP → reverse cash-and-carry

Now suppose the forward is quoted at $103, below fair. Run it backwards:

Short-sell the stock today for $100 and invest the $100 at 6%.
Go long the forward at $103, locking in a purchase price.
At expiry: the investment is worth $106; buy the stock back via the forward for $103 and return it to close the short.
Pocket $3, risk-free.

Buyers swarm the cheap forward, pushing its price up to $106. Between the two arbitrages, the forward is squeezed to its fair value from both sides — which is why, in liquid markets, you essentially never see a forward stray from cost-of-carry.

Think first

In the cash-and-carry arbitrage, the trader is SHORT the forward and LONG the actual stock. Why does this combination carry essentially no price risk?

Hint: Think about what happens to each leg if the stock soars — or crashes — before expiry.

Worked example — a dividend-paying stock

Income the asset pays reduces the forward price, because holding the asset (the carry route) earns you that income, making the carry cheaper. A stock at a spot of $200 (so $S_0 = 200$ ), risk-free rate 5%, one year, pays a $6 dividend during the year. Treating the dividend as a known cash amount $D$ received:

F_0 = (S_0 - \text{PV}(D))(1 + r).

Take the present value of the $6 dividend as roughly $5.71 (discounted at 5%): $F_0 = (200 - 5.71) \times 1.05 \approx 204.00$ , i.e. about $204.00. Compare with the $210 you’d get if the stock paid nothing ( $200 \times 1.05$ ): the $6 of dividends knocked roughly $6 off the forward. The intuition: a forward holder doesn’t receive the dividends (they don’t own the stock yet), so they shouldn’t have to pay for them — the forward price is discounted accordingly.

Two identical stocks trade at $100. Stock A pays no dividend; Stock B pays a generous dividend before the one-year forward expires. With the same interest rate, how do their one-year forward prices compare?

Stock B's forward is HIGHER, because dividends make a stock more valuableThey are identical — dividends never affect forward pricesStock A's forward is lower, because paying no dividend is cheaper to carryStock B's forward is LOWER, because its dividend reduces the net cost of carry

Worked example — a storable commodity and convenience yield

Commodities add two twists: storage costs (which raise the forward) and a convenience yield (which lowers it). Crude oil spot is $80, the one-year rate is 5%, storage and insurance run 3% a year, and the convenience yield — the premium of having actual oil on hand for a refinery that can’t run dry — is 4% a year.

F_0 = S_0\,(1 + r + \text{storage} - \text{convenience}) = 80 \times (1 + 0.05 + 0.03 - 0.04) = 80 \times 1.04 = 83.20.

Net carry is $5\% + 3\% - 4\% = 4\%$ , so the forward is mildly above spot. Crank the convenience yield up to, say, 10% (a genuine shortage — everyone needs oil now) and net carry goes negative: $5\% + 3\% - 10\% = -2\%$ , giving a forward of $80 \times 0.98 = 78.40$ — i.e. $78.40, below spot. That sign flip is precisely the contango→backwardation switch you’ll meet next lesson, and the convenience-yield slider in the chart above is exactly this lever.

Misconception: 'the forward price predicts the future spot price'

The most stubborn confusion in the subject. $F_0 = S_0 e^{(c-y)T}$ contains no forecast — only today’s spot and today’s carry rates. A one-year oil forward of $83.20 is not the market’s bet that oil will be $83.20 in a year; it’s the no-arbitrage price given today’s spot and carry. The actual future spot could be $50 or $120. Forwards price carry, not prophecy. Anyone who reads the forward curve as a price forecast is misreading it.

Match each carry component to its effect on the forward price.

Pick a term, then click its definition.

Fill each blank with the right term — one choice per blank.

Pick the right option for each blank, then check.

The fair forward price equals the price grown by the net cost of . Financing and storage it; income and convenience yield it. If the quoted forward is too high, traders run a arbitrage that pushes it back down. The formula contains of the future spot price — it prices carry, not prophecy.

Putting it together

The fair forward price isn’t a guess and isn’t a forecast — it’s whatever makes “buy forward” cost the same as “buy now and carry it.” That equality gives $F_0 = S_0 e^{(c-y)T}$ : today’s spot, grown by the net cost of carry — financing plus storage, minus income and convenience yield. The formula is enforced by cash-and-carry arbitrage (when the forward is too dear) and its reverse (when too cheap), which let traders extract risk-free profit until the price snaps back. Income and convenience yield drag the forward below spot; financing and storage push it above. And nothing in here predicts the future — it prices carry, not prophecy. Here’s the whole argument on one card:

Big picture

Cost-of-carry pricing — the whole argument

Cost-of-carry pricing
- The core idea
  - Buy forward vs buy-and-carry → same outcome
  - Same outcome must cost the same today
  - That equality IS the formula
- The formula
  - F = S grown by net carry
  - Carry = financing + storage − income − convenience
  - Carry costs raise F; yields lower F
- Enforcement: arbitrage
  - Forward too dear → cash-and-carry, sell it down
  - Forward too cheap → reverse, buy it up
  - Risk-free profit until price snaps to fair
- Special cases
  - Dividends/coupons → forward below pure carry
  - Storage → forward above spot
  - High convenience yield → forward below spot
- The big misconception
  - Forward ≠ forecast of future spot
  - It prices CARRY, not prophecy

Forward = spot grown by net carry, enforced by arbitrage, forecasting nothing.

One mixed recap before we read the whole forward curve:

Question 1 of 50 correct

A non-dividend stock trades at $50. The one-year risk-free rate is 4% (simple annual). What is the fair one-year forward price?

$48.00 — future money is discounted, so the forward is lower$54.00 — spot plus 8% for two years of carry$52.00 — spot grown by 4%: $50 × 1.04$50.00 — the forward always equals spot

Check your answer to continue.

Key Takeaways

What to remember

Forward = spot grown by net cost of carry. $F_0 = S_0 e^{(c-y)T}$ — today’s spot, lifted by financing + storage, lowered by income + convenience yield.
It’s replication, not prediction. “Buy forward” must cost the same as “buy spot and carry it,” because they deliver the identical asset later. That equality fixes the price.
Arbitrage enforces it. A too-expensive forward gets crushed by cash-and-carry; a too-cheap one gets bid up by the reverse. Both extract risk-free profit until the price snaps to fair.
Carry costs raise the forward; holding benefits lower it. Financing and storage push it above spot; dividends, coupons and convenience yield pull it below.
The forward is NOT a forecast. It prices carry, not prophecy — the realised future spot can be anywhere. Misreading the curve as a prediction is the topic’s classic blunder.