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Lesson 1 of 10

What a future is, and fair value

A contract to trade later at a price fixed now, and the no-arbitrage price that fixes it.

A future is an agreement to buy or sell an underlying at a fixed price on a set delivery date. No money changes hands today; the price is locked now, the exchange happens later.

Its fair price is not a forecast. It is pinned by arbitrage: holding the future should cost the same as buying the underlying now and carrying it to delivery. With financing rate \(r\), dividend yield \(q\) and time \(T\):

\[ F = S\,(1 + (r - q)\,T). \]

Strictly, with continuous compounding this is \( F = S\,e^{(r-q)T} \). This trainer uses the linear form \( S(1 + (r-q)T) \), the interview-clean approximation that keeps every number exact; over the horizons here the two agree to a rounding error.

Why it matters
The future is just the spot carried forward. If it trades away from this fair value, a market maker can lock the difference risk-free, so the whole desk is built on this one identity.
Try it

F = S(1 + (r − q)T). The future is the spot carried to delivery.

Fair value
4160
Basis (F − S)
160
Net carry
4%
Worked example

The index is at \(4000\), financing is \(5\%\), the dividend yield is \(1\%\) and delivery is in a year. Fair value is \( 4000\,(1 + (0.05 - 0.01)\times 1) = 4000 \times 1.04 = 4160 \).

In the interview
What they are testing
That you can write down cost-of-carry fair value and treat the future as carried spot.
How to narrate it
State the identity, then plug in: "fair value is spot times one plus net carry times time, so 2000 times 1.02 is 2040."
Common mistake
Treating the future as a forecast of where the index will be, or dropping the time factor T.
Quick check

Index at 2000, r = 6%, q = 2%, T = 0.5. What is the fair value of the future?