A portfolio manager runs a $500 million equity fund and wishes to hedge her stock portfolio over

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Question:

A portfolio manager runs a $500 million equity fund and wishes to hedge her stock portfolio over the final one and a half months of the year. The beta of her stock portfolio is 1.45. Her portfolio has performed quite well relative to comparable funds over the first ten and a half months of the year and she now wants to hedge against a possible crash in the stock market during the final stretch of the year. Rather than liquidating her stock portfolio or purchasing put options, she wants to use stock index futures to hedge her portfolio. Although the (maturity matched) one and a half-month futures contract is not available, a three-month futures contract is traded.

Each futures contract is for delivery of $250 times the index. The stock index currently stands at 21,000 and its dividend yield is 2.5%. The risk-free rate is 1.25%. The manager does not want to offload all market risk but does wish to lower the beta of her portfolio to 0.45.

a) What position in the three-month futures contract should she take?

b) If the stock index drops to 20,150 by yearend, how does the hedge work out? Assume there is no
change in either the risk-free rate or the dividend yield on the stock index.