Suppose a fund manager has a portfolio of assets that is worth $60 million with a beta of
0.75 with respect to the Nasdaq index. To be simple, assume that the alpha and expense ratio
of the portfolio are both zero. The manager is concerned about the performance of the Nasdaq
market over the next three months and plans to use six-month mini futures contracts on the
Nasdaq 100 to hedge the risk. The current level of the index is 5914. One contract of mini
Nasdaq 100 futures is on $20 times the index. The risk-free rate is r =0.65% per annum. The
dividend yield on the Nasdaq index is y =1.12% per annum.
(a) What is the theoretical futures price for the six-month mini Nasdaq 100 futures?
(b) If the fund manager ignores the mismatch between the futures maturity and the hedging
period and simply uses the formula we discussed in class, how many futures contracts
(round the number to an integer) will he long or short for the purpose of minimizing the
exposure to the market over the next three months?
(c) How much does the fund manager gain (lose) on his futures position if the market index
in three months turns out to be 4731, 5323, 5914, 6505, and 7097? List the profits and
losses (P/L) corresponding each index level in a table.
(d) Suppose there are no idiosyncratic shocks to the portfolio returns. (This is true if the
portfolio only consists of the index and three-month Treasury bills.) Calculate and report
the P/L of the portfolio when the index finishes in three months at the five levels listed
in question (b). To simplify the calculation, you can approximate the simple k-month
dividend yield by y ×(k/12), i.e., ignore the effect of continuous compounding. Similarly,
approximate the k-month simple interest rate by r ×(k/12).
(e) Calculate the P/L and return on the hedged portfolio when the index in three months
equals the five levels listed in question (b).
(f) Explain why the hedge is not perfect even when there are no idiosyncratic shocks to the
portfolio returns.

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