The following examples are quick intuitive approximations for fair value relationships for the Stock Market Indices such as the S&P 500.

Again, key determinants of calculating fair value are:

**CI** = Cash Stock Index
**FP** = Futures Price
**i** = interest rate or discounting factor
**D**_{pv} = expected value of dividends during holding period
**t** = time between now for cash and then for futures maturity or expiration
These factors are constructed into the following mathematical expression:

**FP = CI (1 + i)**^{t} - D_{pv}
The first example assumes that dividends are equal to zero (D_{pv} = 0). In this case the expression simply imputes a higher price for the futures over cash or actual stocks.

Also, assume that **i** is equal to 5 percent per annum and the cash index is 1,000. If the time horizon is 1 year, the implied futures price is 1050, or futures are 50 over cash.

FP = 1000(1 + .05)^{1} - 0 {zero for no dividends}

or,
FP = 1000(1.05)^{1} which solves to
FP = 1050
Quickly determining the quarterly fair value difference entails dividing the annual 50 point spread or difference by 4. This results in a theoretical difference of 12.50 points, futures over the cash index.

Therefore, if futures were trading at a spread less than 12.50 points to the cash index (basket of actual stocks), then an arbitrageur would buy the futures and sell the actual stocks until the relationship between the 2 components equalled 12.50 points.

Conversely, if the futures were trading at 20 points over cash, then it would be expected that arbitrageurs would sell futures and buy stocks until balance was restored at the spread of 12.50 points. In the later example, there was a theoretical profit opportunity equal to 7.50 points, basis the futures.

Expressed another way, when an announcer on a television show such as CNBC states that futures are now trading 7.50 points above **FAIR VALUE**, the reporter means that stocks can be expected to trade higher in the very near-term. This expected rise in actual stocks would reduce the prevailing premium to the fair value premium level.

When interest rates are 6 percent, in the absence of dividends, one would expect one year futures to have a 60 point premium relative to cash. The other simple estimates would be 15 points for a 3 month contract, 30 points for a 6 month contract and 45 points for a 9 month contract.

When dividends are considered, a 2 percent yield (payable one year hence) would suggest reducing the futures value for the one year contract by 20 points.

The art and science of these programs fine tunes the actual expected payment streams and incorporates adjustments for trading costs.

Often these relationships reflect the cost structure of the most efficient participants. That is, if the most efficient participant can borrow at LIBOR but another participant must borrow at LIBOR plus 100bps, then the former has an advantage and will drive the spreads to tighter differences. Returning to the above examples, one investor has a finance rate of 5 percent whereas a competitor has to pay 6 percent (LIBOR plus 100 bps.) It is shown that the first trader has a 10 full point comparative advantage.

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