Futures Fair Value, Basis & Roll Calculator

About the Futures Fair Value, Basis & Roll Calculator

A future is worth what it costs to buy the asset now and hold it until expiry: the spot price grown at the interest rate, less the dividends the holder of the asset would receive, plus storage costs, less any convenience yield. This cost-of-carry fair value is where arbitrage keeps a future close to spot.

Type the spot price, the days to expiry and the interest rate (simple or continuous, ACT/365 or ACT/360), and a dividend yield or the dividends themselves. Add the futures price you see quoted and the calculator shows the basis (futures − spot), the premium, the interest rate the price implies and how far the market is from fair value. For a roll, enter the near and far prices: it works out the roll cost and the annualised roll yield, and says whether the prices you entered are in contango or backwardation.

It works for any index, commodity or bond future in any market, from the numbers you type: no market data is used.

Common uses

• Check whether an index future’s premium over spot matches the interest rate you can earn.
• Work out the fair value of a future around a large dividend.
• See the cost of rolling a futures position into the next expiry, and its annualised yield.
• Compare the rate implied by a futures price with a deposit or repo rate.

The formulas

• Simple interest: FV = S × (1 + c × d ÷ B) − Σ D × (1 + r × (d − t) ÷ B), where c = r − q + u − y.
• Continuous: FV = (S − Σ D × e^{−r t ÷ B}) × e^{c × d ÷ B}.
• Implied rate: simple (F ÷ S − 1) × B ÷ d + q − u + y; continuous ln(F ÷ S) ÷ T + q − u + y. Each undoes its own fair value formula, so both are the interest rate that makes the quoted price fair.
• Roll cost = (F far − F near) × multiplier × contracts; annualised roll yield = (F far ÷ F near − 1) × 365 ÷ days, not shown for rolls under 7 days apart.

S is spot, d the days to expiry, B the day count (365 or 360), r the interest rate, q the dividend yield, u the storage cost, y the convenience yield, D a dividend paid t days from now and F a futures price. These are the cost-of-carry relations in Hull, Options, Futures, and Other Derivatives, chapter 5 (F₀ = S₀e^{(r − q)T}, F₀ = (S₀ − I)e^{rT} with I the present value of the income, and storage and convenience yields for commodities).

Reading the results

• Fair value − spot is the cost of carry: positive when interest outweighs the income, negative when dividends or a convenience yield outweigh it.
• Basis here is futures − spot; some markets quote it the other way round.
• The rate the price implies is the interest rate that would make the quoted futures price fair; compare it with the rate you can borrow or lend at.
• Contango and backwardation describe only the prices you entered: a later price above an earlier one, or below it.

What it leaves out

Transaction costs, the bid–ask spread, taxes, the difference between borrowing and lending rates, margin and daily settlement, and the delivery options of bond and commodity futures (which make their fair value lower than this simple model says). Index dividends are estimates until they are paid.

Limitations

• One flat interest rate for the whole period; real rates follow a curve.
• Storage costs and convenience yields are taken as yearly percentages of the price.
• The implied rates compare the futures price with spot and add back the dividend yield, storage cost and convenience yield you enter; dividends you list one by one are left out of them.
• A roll yield is scaled up to a year from the days between the expiries: over a short gap it can look much larger than the cost really is, and under 7 days it is not shown.
• It calculates from your numbers and uses no market data: it says nothing about where any price is going.

Privacy

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