A pension fund is currently valued at $250 million has a beta of 0.8 and an alpha of
0.5% annual. The fund's annual volatility is 12.5% and the annual volatility of the
market index return is 15.0%. There exist index futures with contract size of $50 per
index point. The stock index is currently at 550 and the index pays 4.0% of dividends
annually. The annual interest rate is constant at 5.0% per year.

(a) How can the pension fund manager decrease the beta of the fund to 0.5 over the
next year and what is the exact position that needs to be taken?
(b) How much is the idiosyncratic risk (risk uncorrelated with the market) of the fund
before and after the position that needs to be taken in (a)?
(c) Suppose that $50 million of new money enters into the fund and the fund manager
wants to create a synthetic investment on the market index of the same amount.
What needs to be done and how is the strategy called?
(d) Suppose that $50 million of new money enters into the fund and the fund manager
wants to create a synthetic investment on the market index but so that the beta
of the fund remains the same and equal to 0.8. What positions need to be taken
and what will the new alpha and idiosyncratic risk be?