Consider a passive mutual fund, an active mutual fund, and a hedge fund. The mutual funds claim to deliver the following gross returns: r passive fund before fees stock index =rt active fund before fees = 2.20%+rstock index + Et The stock index has a volatility of var (r stock index) = 15%. The active fund has idiosyncratic risk of var(+) = 3.5%. The passive fund charges an annual fee of 0.10% and the active mutual fund charges a fee of 1.20%. The hedge fund uses the same strategy as the active mutual fund to identify "good" and "bad" stocks but implements the strategy as a long-short hedge fund, applying 4 times leverage. The risk-free interest rate is r = 1% and the financing spread is zero (meaning that borrowing and lending rates are equal). Therefore, the hedge fund's return before fees is hedge fund before fees rt = 1% +4 (ra - _active fund before fees passive fund before fees) Answer the following questions, and explain your answers thoroughly: a) What is the hedge fund's volatility? (10p) b) What is the hedge fund's beta? (10p) c) What is the hedge fund's alpha before fees? (20p) d) Suppose that an investor has $40 invested in the active fund and $60 in cash. What investments in the passive fund, the hedge fund, and cash (i.e., the risk-free asset) would yield the same market exposure, same volatility, and same exposure to idiosyncratic risk et? (20p) e) Based on your answer above, what is the fair management fee for the hedge fund in the sense that it would make the investor indifferent between the two allocations (if the hedge fund charges a performance fee of zero)? (20p) f) If the hedge fund charges a management fee of 2%, what performance fee makes the expected fee the same as above? Ignore high water marks and ignore the fact that returns can be negative but recall that performance fees are charged as a percentage of the (excess) return after management fees. Specifically, assume the performance fee is a fraction of the hedge fund's outperformance above the risk-free interest rate. (20p)