CFA Level IDerivativesMedium
A stock index currently trades at 4,000. The one-year risk-free rate is 5%, and the index is expected to pay a dividend yield of 2% over the year. Using the cost-of-carry model with discrete compounding, what is the no-arbitrage one-year futures price on the index?
- A$4,117.65
- B$4,000.00
- C$4,200.00
- D$3,882.35
Show answer & explanationAnswer & explanation
Correct answer: A. $4,117.65
F0(T) = S0 × [(1+Rf)/(1+δ)]^T = 4,000 × (1.05/1.02)^1 = 4,000 × 1.029412 = $4,117.65. The futures price incorporates the risk-free rate minus the dividend yield as the net cost of carry.
Why the other options are wrong
- B. This assumes zero net cost of carry, ignoring the rate differential.
- C. This overstates the adjustment by using a larger implied carry cost.
- D. This inverts the ratio (uses δ over Rf) instead of Rf over δ.
Futures Price with Dividend Yield (Cost of Carry)
The no-arbitrage futures price on a dividend-paying index equals the spot price adjusted for the risk-free rate net of the dividend yield.
- F0(T) = S0 × [(1+Rf)/(1+δ)]^T
- Higher dividend yield lowers the futures price relative to spot
- Higher risk-free rate raises the futures price relative to spot
Memory trick: Carry cost = borrow rate minus dividend gain — net it into the futures price.