2.22 Suppose you have some money to invest—for simplicity, $1—and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that $1 invested in a stock fund yields Rs after 1 year and that $1 invested in a bond fund yields Rb, suppose that Rs is random with mean 0.08 (8%) and standard deviation 0.07, and suppose that Rb is random with mean 0.05 (5%) and standard deviation 0.04. The correlation between Rs and Rb is 0.25. If you place a fraction w of your money in the stock fund and the rest, 1 - w, in the bond fund, then the return on your investment is R = wRs + (1 - w)Rb.

a. Suppose that w = 0.5. Compute the mean and standard deviation of R.

b. Suppose that w = 0.75. Compute the mean and standard deviation of R.

c. What value of w makes the mean of R as large as possible? What is the standard deviation of R for this value of w?

d. (Harder) What is the value of w that minimizes the standard deviation of R? (Show using a graph, algebra, or calculus.)