6. Suppose Ms. X has 900 shares of PQR stock and Mr. Y has 600 shares of XYZ stock. At t=0 P_PQR = $100 and P_XYZ = $150. Ms. X and Mr. Y decide to get married, and want to allocate their combined wealth between the two stocks in such a way that the standard deviation of the returns on their portfolio is minimized. You are given that (i) annualized expected return on PQR stock is 12.6% and that on XYZ stock is 13.5%, (ii) the variance of annualized returns on PQR stock is 0.09 and that on XYZ stock is 0.16, and (iii) the correlation between the returns on the two stocks is 0.27.

[1 point each for parts (a) and (b), 2 points each for parts (c) to (g) for a total of 12 points]

(a): What is the value of (i) Ms. Xs and (ii) Mr. Ys stock holdings?

(b): What is the current portfolio composition? That is, denoting PQR as Security 1, and XYZ as Security 2, what are the current fractions invested in the two stocks in the combined portfolio?

(c): Work out portfolio composition of the risk minimizing portfolio? What are the portfolio weights? What is the dollar amount invested in PQR and in XYZ?

(d): Work out the number of shares in each of the two stocks that the risk minimizing portfolio will contain.

(e): Work out the expected value of their portfolio at t=1 (one year from today)

(f): Work out the range within which their t=1 wealth will lie with a probability of 95% assuming that the future wealth is normally distributed.[1]

(g): Check that your answer for the risk minimizing portfolio weights does in fact minimize the risk with the help of the MS Excel file 2-Security Portfolios.xlsx on the course web site by entering your solution in an appropriate row in that file, and showing that the portfolio risk increases if the portfolio weights differ from the optimal by 0.0001 in either direction. Submit the Excel file with your exam.

[1] The lower end of the 0.95 confidence interval is [E(Wealth₁) 2*SD(Wealth₁)] and the upper end is [E(Wealth₁) + 2*SD(Wealth₁)].