How do you calculate the fair value of an equity index future and identify when it trades rich or cheap?

The fair value of an equity index futures contract is determined by the cost-of-carry model, which balances the financing cost of holding the index against the dividends received.

Cost-of-Carry Formula (Continuous)

F = S x e^{(r - q) x T}

where:

• S = current spot index level
• r = risk-free rate (continuously compounded)
• q = continuous dividend yield
• T = time to expiry in years

Worked Example

The Grandview 500 index is currently at 5,280. The 3-month risk-free rate is 4.60% (continuously compounded), and the index has a continuous dividend yield of 1.45%. The futures contract expires in 63 trading days (roughly 91 calendar days).

T = 91 / 365 = 0.2493

F = 5,280 x e^{(0.046 - 0.0145) x 0.2493}

F = 5,280 x e^{0.00785}

F = 5,280 x 1.00788

F = 5,321.6

If the actual futures price is 5,340, the contract is trading 18.4 points rich to fair value.

Arbitrage Implications

Practical Frictions

In reality, transaction costs, margin requirements, short-selling constraints, and the discrete nature of dividends create a no-arbitrage band around fair value — typically 1-3 index points for major indices.

Exam Tip: If the question gives discrete dividends instead of a continuous yield, use F = (S - PV(Dividends)) x e^{r x T} instead.

Check out our FRM Part I question bank for more futures pricing problems.