Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 10

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 R s after 1 year and that $1 invested in a bond fund yields R b , suppose that R s is random with mean 0.08 (8%) and standard deviation 0.07, and suppose that R b is random with mean 0.05 (5%) and standard deviation 0.04. The correlation between R s and R b 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 = wR s + (1 w ) R b.
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.)