Futures contracts are sometimes used to alter the beta of a portfolio which has been reduced to zero (in order to insulate expected return from the index performance. Suppose, for example, that the S&P500 index is currently trading at 1500, with the index futures price being set at 1515. For a portfolio worth $10 million, with a beta of 1.2, if each futures is on $250 times the index (or the contract size), then the current value of one futures contract is:

F = futures price x contract size

F = 1515 x 250

F = 378,750

Number of futures to be short = beta x (portfolio value/ futures value)

Number of futures to be short = 1.2 x ($10,000,000/ 378,750)

Number of futures to be short = 32

In order to decrease beta, a position should be taken in:

Number of short futures = (old beta- new beta) x portfolio value/ futures value

On the contrary, to increase beta, a position needs to be taken in:

Number of long futures = (new beta- old beta) x portfolio value/ futures value

In our example, to reduce the beta of that portfolio from 1.2 to 0.8, the number of contract shorted should be:

32 x (0.8/ 1.2) = 22 futures contracts