COMM 371 Practice Final Exam Two problems

Problem 1.Assume that the risk free return is 1%, the S&P500 expected return is 12% and the S&P500 volatility is 25% (for a one year horizon). Suppose that you are a pension fund manager and that your investment goal is to have an expected return of 15% (for a one year horizon).(d) Suppose a hedge fund can deliver a return that has a beta of 1.1 and a standard deviation of30%. Also assume that you are convinced that the hedge fund can generate an alpha of 3%.Based on this you decided to give the hedge fund a weight of 20%. What weights should youchoose for the risk free security and the S&P500, so that your expected return is still 15%?Answer: It is easy to check that the weights should be 1 in the S&P 500 (more specifically0.9981) and -0.2 in the bond (more specifically -0.19818).Problem 2 (Portfolio choice)You decided to invest $200M in the stocks of two companies: A and B. The expected returns and standard deviations of returns for the two stocks are as follows:StocksABExpected return8%13%Standard deviation12%20%(a) What must be the correlation of the return of stock A and the return of stock B if you must invest$164M in Stock A in order to form the global minimum variance portfolio (among all portfolios thatcombine stock A and stock B)?Answer: We see that w=0.82 and we invert the the formula22 12 1 2w1 = 221 + 2 2 12 1 2and obtain rho= 0.3.w= 0.82 in Stock A (we invest $164M). The expected return is 8%*0.82 + 13%*0.12 =8.9% and the the stdv is 11.45%) (apply the formulae of the variance).(b) Assume now that, in addition to stock A and stock B, it is possible to invest in a risk free securitywith an interest rate of Rf =5%. The resulting tangency portfolio has an expected return of 11%.Assume again that you have $200M and that you want the highest expected return, provided that thestandard deviation of your portfolio is 12%. Compute the amounts that you should invest is stock A,stock B and the risk free security as well as the resulting expected return.Answer: First, we find the weight of the tangency portfoliofrom the expected return formula and compute the resulting stdev (14.2%).You must invest 16% ($32M) in the risk free asset and 84% ($168M) in the tangency portfolio meaning that you have to invest16% in the risk free asset, 50.4% ($100.8M) in A and33.6% ($67.2M) in B. The resulting expected return is 10.4% (higher than 8%).(c) Briefly sketch a risk return (,Er) diagram positioning the risk free security, stock A, stock B, thetangency portfolio, the portfolio formed in question (a) and the portfolio formed in question (b).1 and P=7.21