The risk-free rate of interest is 7 % per annum with...

The risk-free rate of interest is $7 \%$ per annum with continuous compounding and the dividend yield on a stock index is $3.2 \%$ per annum. The current value of an index is 150. What is the 6-month futures price?

Key Concepts

Continuous Compounding

Continuous compounding is a method of calculating interest whereby the interest is added to the principal at every possible instant, leading to exponential growth. This approach simplifies mathematical modeling in finance by using the exponential function to represent accrued interest over time.

Futures Contracts

A futures contract is a standardized agreement to buy or sell an asset at a specific future date for a predetermined price. It is used for hedging, speculation, and arbitrage, with its pricing reflecting the cost-of-carry of holding the underlying asset over time.

Cost-of-Carry Model

The cost-of-carry model is used to price futures contracts by relating the current spot price to the futures price through the net cost of holding the asset. This model typically incorporates the financing cost (such as the risk-free rate) and any benefits from holding the asset (such as dividend yield), leading to an exponential adjustment in continuous compounding scenarios.

Risk-Free Rate

The risk-free rate represents the theoretical return on an investment with no risk of loss, typically derived from government bonds. It is used as the baseline rate in pricing models to measure the opportunity cost of investing capital, and it plays a central role in adjusting the spot price when calculating futures prices.

Dividend Yield

Dividend yield is the annual dividend income expressed as a percentage of the asset's price. In the context of futures pricing, the dividend yield reduces the effective cost of carrying the asset because it represents income that the holder would receive, thereby reducing the net cost-of-carry.

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