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 one 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 \(R=w R_{s}+(1-w) R_{b}\).

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.