On January 31st 2020, you have a portfolio of the IJR small cap ETF only, worth $ 980,000 (at the end of trading day)

You would like to partially hedge the portfolio with S&P 500 index futures contracts. Here we ignore the risk-free rate or assume it is equal to zero.

a. Compute the initial beta of the portfolio with respect to the S&P 500, using regular returns (P2/P1-1) computed from prices available from January 2nd 2020 to January 31st 2020.

b. Determine the optimal number of index futures contracts needed (short position) in order to reduce the beta to a value of 0.3

Note that the size of an S&P 500 index futures contract is 250 times the futures price (S&P 500 futures price on January 31st 2020.)

c. What is the return on your portfolio WITHOUT the futures contracts between January 31st and February 28th? What is the return of the S&P 500 for the same period?

d. Compute the dollar loss on your portfolio, as well as the dollar gain on the short index futures position. What is your net gain/loss in total?

e. Calculate the theoretical percentage rate of return you should have lost if the total portfolio had indeed a beta of 0.3, i.e. multiply the percentage loss in the S&P 500 by 0.3

f. Compare your answer in part e) to your actual percentage loss in total, i.e. the net total loss found in part d) divided by the initial (January 31st) portfolio value.

If they are close (percentage theoretical loss and percentage actual loss being less than 0.1% apart), it means that you have succeeded at reducing your initial beta to a beta of 0.3