Forwards, futures and the cost of carry

A forward price is not a forecast. It is the spot price plus the cost of holding the asset until delivery, and it is enforced by a trade anyone can put on. Getting this straight removes most of the confusion around contango, backwardation and what a futures curve is telling you.

Equation 1.1

The no-arbitrage forward

Spot, less any income the asset throws off, carried forward at the financing rate.

F_0 = (S_0 - D)e^{rT}

$S_0$: Spot price today.
$D$: Present value of income received while holding — dividends, coupons.
$r$: Financing rate; for commodities add storage and subtract convenience yield.

Derivation 1.2

Cash and carry

The formula is enforced by a portfolio anyone can build today.

$\text{Borrow } S_0,\ \text{buy the asset, sell the forward}$
You now own the asset, owe the loan, and are committed to deliver at $F_0$ .
$\text{At } T:\ \text{deliver for } F_0,\ \text{repay } S_0 e^{rT},\ \text{having collected } D e^{rT}$
$\text{Profit} = F_0 - (S_0 - D)e^{rT}$
Riskless, and known today. So it must be zero, or everyone does it.

F_0 = (S_0 - D)e^{rT}

Why the forward is not a forecast. If a forward were priced at anyone’s expectation of the future spot, the cash-and-carry trade would print money whenever that expectation differed from spot plus carry. The forward sits where it does because of what it costs to hold the thing, not because of where anyone thinks the price is going — which is why a steep curve is a statement about rates and storage, not about direction.

Shape	Means	Typical cause
Contango: $F > S$	Carry is positive	Financing and storage exceed any income or convenience yield
Backwardation: $F < S$	Carry is negative	Large dividends, high convenience yield, or a shortage of the physical

Table 1.3 · Contango and backwardation. Rolling a long position in contango loses money as each contract converges down to spot — the reason leveraged commodity products decay.

Example 1.4

An index is at $4{,}000$ . Rates are $5\%$ and the dividend yield is $2\%$ , both continuously compounded. What is the six-month forward?

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Worked solution

Formula $F_0 = S_0 e^{(r - q)T}$
Substitute $= 4000\, e^{(0.05 - 0.02)(0.5)}$
Solve $= 4000\, e^{0.015}$
$\approx 4000 \times 1.01511$
Answer $F_0 \approx 4{,}060$

Sanity check. Above spot, because financing costs more than the dividends pay. If the yield exceeded the rate the forward would sit below spot, with no change in anyone’s view of the index.

Proposition 1.5

Where futures actually differ

A future is margined daily, so gains and losses are paid in cash as they happen rather than at delivery. That creates a correlation effect: if the asset tends to rise when rates rise, a long future receives cash exactly when it can be reinvested well, which makes the future worth slightly more than the forward. With deterministic rates the two are identical.

Holds when

For short-dated equity contracts the difference is negligible and nobody adjusts for it.
For long-dated interest-rate futures it is material, and the convexity adjustment is a real number on a real desk.

Definition 1.6

Basis and convergence

Basis, $\text{basis} = S_t - F_t$ — The gap between spot and the futures price. It narrows mechanically as delivery approaches, because the carry left to pay shrinks, and it is zero at expiry — which is what makes the cash-and-carry trade close cleanly.

Common trap. Reading an upward-sloping futures curve as the market predicting higher prices. Contango is usually just positive carry, and an index forward above spot says only that rates exceed the dividend yield. Instead. Ask what it costs to hold the asset. If the curve is steeper than carry explains, *then* you have found something worth discussing.

A forward price is a cost, not a forecast. The forward is where you can lock a price today by borrowing, buying and storing — so it is set by financing and carry, not by anyone’s view. If the forward were a forecast, a stock everyone expected to double would trade at a huge forward premium; it does not, because the arbitrage does not care what you expect.

Example 1.7

A one-year forward

A non-dividend stock is at $\$200$ and rates are $4\%$ continuously compounded. Where is the one-year forward, and what if the stock pays a $2\%$ continuous dividend yield?

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Worked solution

Formula $F = S_0 e^{(r-q)T}$
Substitute $200 e^{0.04}, \quad 200 e^{0.02}$
Solve $200 \times 1.040811 = 208.16$ No dividend: pure financing.
$200 \times 1.020201 = 204.04$ The yield you collect offsets the rate you pay.
Answer $\$208.16 \text{ and } \$204.04$

Sanity check. The dividend halves the carry because it halves the net cost of holding the share — and when $q > r$ the forward sits below spot, which is backwardation without any view attached.

Example 1.8

The basis and what it must do

The same stock is at $\$200$ with the one-year future at $\$212$ and rates at $4\%$ , no dividend. Is there a trade?

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Worked solution

Formula $\text{basis} = F_{\text{market}} - S_0 e^{rT}$
Substitute $212 - 208.16$
Solve $= 3.84$ The future is rich to fair.
$\text{Sell the future, buy the stock, borrow } 200$ A cash-and-carry.
Answer $\$3.84 \text{ locked, whatever the stock does}$

Sanity check. The basis must go to zero at expiry because the future settles into the spot, so the profit is not a view on convergence — it is arithmetic that completes itself.

Equation 1.9

Commodities: storage and convenience yield

Holding a physical commodity costs financing $r$ and storage $u$ but earns a convenience yield $y$ — the benefit of having it on hand when supply is tight. When $y$ exceeds $r + u$ the curve is backwardated. Unlike a financial asset, $y$ is not observable directly: it is backed out of the curve.

F_0 = S_0\,e^{(r + u - y)T}

$u$: Storage cost as a continuous rate.
$y$: Convenience yield, implied by the futures curve.

Example 1.10

The implied financing rate

A non-dividend stock trades at $\$100$ and its six-month future at $\$102$ . What continuously compounded financing rate does the future imply?

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Worked solution

Formula $r_{\text{impl}} = \frac{1}{T}\ln\frac{F}{S}$
Substitute $\frac{1}{0.5}\ln\frac{102}{100}$
Solve $2 \times 0.0198$
Answer $\approx 3.96\%$

Sanity check. If you can borrow for less than $3.96\%$ , buying stock and selling the future locks in the difference. Futures desks quote this implied repo rate constantly, because it is the cheapest-to-finance comparison across contracts.

Example 1.11

Daily variation margin

You are long $10$ E-mini S&P 500 contracts (multiplier $\$50$ ). The settlement price falls from $5{,}000$ to $4{,}950$ . What variation margin do you pay?

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Worked solution

Formula $\text{VM} = n \times m \times \Delta F$
Substitute $10 \times 50 \times (-50)$
Solve $= -25{,}000$
Answer $\$25{,}000 \text{ paid}$

Sanity check. Paid in cash that day, not at expiry. A forward would record the same loss but settle it later — the whole difference between the two instruments is when the cash moves.

Equation 1.12

Rate futures versus forwards

Because a rate future is margined daily, and gains are received when rates are high and can be reinvested at high rates, a short rate-future position is worth more than the equivalent forward. So futures-implied rates sit above forward rates, by an amount that grows with volatility and roughly with the square of maturity.

f_{\text{fwd}} \approx f_{\text{fut}} - \tfrac{1}{2}\sigma^2 T_1 T_2

$\sigma$: Normal volatility of the short rate.
$T_1, T_2$: Start and end of the rate period.

Example 1.13

Sizing the convexity adjustment

A rate future covers the period from $5$ to $5.25$ years, and short-rate volatility is $1\%$ a year (normal). By roughly how much does the futures rate exceed the forward rate?

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Worked solution

Formula $\tfrac{1}{2}\sigma^2 T_1 T_2$
Substitute $\tfrac{1}{2} \times 0.01^2 \times 5 \times 5.25$
Solve $= 0.00131$
Answer $\approx 13\ \text{bp}$

Sanity check. Negligible for the front contracts and material for long-dated ones — the reason curve builders adjust futures before bootstrapping swap curves out to many years.

The reverse trade needs a borrow. If a forward is too cheap, the arbitrage is to sell the asset short, lend the proceeds and buy the forward. That requires borrowing the asset — possible for liquid stocks and bonds, often costly for hard-to-borrow names, and impossible for most physical commodities. Which is why forwards on hard-to-borrow stocks can sit below the textbook price for long periods.

Common trap — forgetting the income. Pricing an index or stock forward as $S e^{rT}$ when the asset pays dividends. The forward holder does not receive them, so the forward is lower by their value. Instead. Subtract the present value of known dividends from spot, or use $e^{(r - q)T}$ with a continuous yield. For single stocks around a large dividend, the discrete form matters.

What you need to know

$F_0 = (S_0 - D)e^{rT}$ , enforced by cash and carry rather than by anyone’s forecast.
For an index with a continuous yield, $F_0 = S_0 e^{(r-q)T}$ .
Contango and backwardation describe the sign of carry, not a market view.
Futures differ from forwards only through daily margining, which matters when rates are stochastic.
Basis converges to zero at delivery.

Exercise 1.14

A stock pays a large dividend just before a forward’s delivery date. Does the forward trade above or below spot, and what does that say about the market’s view?

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Below spot, if the dividend exceeds the financing cost — you receive the dividend while holding the stock, so the forward buyer must be compensated for missing it. It says nothing whatever about anyone’s view; it is pure carry.

Exercise 1.15

An implied dividend yield

An index is at $100$ and its one-year forward at $101$ , with rates at $4\%$ continuously compounded. What dividend yield does the forward imply?

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$q = r - \ln(F/S) = 4\% - \ln 1.01 = 4\% - 0.995\% \approx 3.0\%$ . Dividend traders back out implied dividends this way and trade them against their forecasts.

Exercise 1.16

A short future

You are short $4$ contracts with a $\$1{,}000$ multiplier and the settlement price rises $2.5$ . What happens to your margin account?

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You pay $4 \times 1000 \times 2.5 = \$10{,}000$ of variation margin that day.

In the interview

Derive it with the cash-and-carry trade rather than quoting the formula. Saying "borrow, buy, sell the forward — that has to be worth nothing" answers the question and pre-empts the follow-ups about dividends, storage and borrow, all of which are just extra terms in the same argument.

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After this lesson you should be able to

What you need to know

In the interview