Statistics are central to managing an investment portfolio. Imagine that an investor is choosing how to divide each dollar between two investments,...

Statistics are central to managing an investment portfolio. Imagine that an investor is choosing how to divide each dollar between two investments, labelled A and B. Investment A has a return with mean 2 and standard deviation 1 and so does investment B. The correlation between the two returns is 0.5. The portfolio invests a fraction ω in investment A and a fraction 1 − ω in investment B so that the portfolio's return is:

rp = ωrA + (1 − ω)rB.

(a) Find formulas for the mean of the portfolio return, labelled µp, and the variance of the portfolio return, labelled σ 2 p , as functions of ω. Do the mean and variance of the portfolio return depend on the value of ω?

(b) In finance, risk somestimes is measured using the variance. Find the value of ω that minimizes the variance of rp. (Hint: Use your mathematics skills to minimize the function you derived.)