The fair value premium for a stock index futures contract is defined simply as the theoretical futures price minus the cash index price. The fair value premium shows how far the futures contract should be trading above or below the cash index given expected dividend income for the stocks in the S&P 500, the days to expiration for the futures contract and the short-term interest rate. If the future's price moves too far above or below the fair value premium band, then one can expect index arbitragers to begin executing trades which will bring the cash and the futures prices back into line.

In order to calculate the fair value premium, it is first necessary to calculate the theoretical S&P 500 futures price. Optima uses the standard theoretical futures valuation formula to compute the S&P 500 theoretical futures price. This formula is explained in detail in the Chicago Board of Trade's pamphlet entitled CBOT Financial Update or in the Journal of Futures Markets, (Jan 1983, Vol 3, No 1), in an article entitled "The Pricing of Stock Index Futures", by Bradford Cornell and Kenneth R. French.

The formula is as follows:      F = (s * e^rt) - D

where:

• F = Theoretical Futures Price

• s = Spot Index Price

• e = 2.7183 which is the Natural Exponential Function

• r = short-term interest rate

• t = number of days to futures expiration/360

• D = dividends, expressed in index points, for stocks going ex-dividend prior to futures expiration.

For example, assume:

-  a cash S&P 500 index price of 352.00
-  a short-term interest rate of 4.50%
-  65 days until futures contract expiration
-  dividend income of 2.17738 S&P 500 index points through the futures contract expiration date.

Using that data the theoretical futures price is calculated as follows:

Theo. futures price = [(352.00)(2.7182)^{(.0450x65/360)}] - (2.17738) = 352.69 points

The fair value premium is then calculated as follows:

Fair value premium = Theoretical Futures Price of 352.69 minus Cash Price of 352.00 = 0.69 points.

If strong buying in the futures market pushes the futures prices higher relative to the cash index so that the actual premium moves to 2.19 points, for example, which is 1.50 points above the fair value premium of 0.69 points, one would expect arbitragers to initiate buy programs in which they buy the cash stocks and sell equal amounts of the futures. This type of index arbitrage performs the function of keeping the futures and cash prices in line and provides liquidity to the markets.

The S&P fair value premium is computed on a daily basis for the front three Chicago Mercantile Exchange S&P futures contracts and the front three Chicago Board of Trade Dow Industrial futures contracts. Three different interest rate assumptions are used for comparison purposes:  the Eurodollar rate, the T-bill rate, and the average of the Eurodollar and T-bill rates. These three different rates to allow the user to see how different rates affect the premium.

The premiums calculations use a a data base of the daily dividend payments expected on an S&P 500 stock portfolio. This data base is updated periodically and is used to derive the value of dividends for stocks going ex-dividend prior to expiration.