12 OPTIMUM ALLOCATION OF INVESTMENTS The joint distribution of the rates of return (in %) of an income fund (X) and a money market fund (Y) is given as follows: XY 5 7 9 11 5 .04 .06 .10 .20 6 .06 .08 .10 .06 7 0 .06 .10 .04 8 0 0 0 .10 1. Show that 2. Let P be the proportion invested in the income fund and let 1-P be the proportion invested in the money market fund. Express the expected rate of return and the variance as a function of P. Tabulate these values for P = 0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0. 3. What proportion should be invested in each fund in order to maximize the expected rate of return? 4. What proportion should be invested in each fund in order to minimize the risk (variance)? Try doing this problem by inspection of the table provided in problem 2, and by using differential calculus. 5. Is there a benefit to diversification with a positive correlation? Discuss. 6. Suppose you wish to choose the portfolio which will provide the highest expected rate of return, however, you want your portfolio to be no more risky than a money market fund. How should your capital be allocated? (Assume that the variance is your measure of risk.) 7. Suppose the joint distribution of the rates of return on the income (X) and money market fund (Y) is bivariate normal with X 9, Y 6, X 2, Y 1 and .12. X 9, Y 6, X 2, Y 1 and .12. Further, you decide to invest 20% of your funds in the income fund and 80% in a money market. 104 a. Compute the mean rate return on your portfolio. b. Compute the standard deviation of the rate of return. c. Compute the probability that the rate of return on your portfolio will exceed 6%. 8. Suppose in another investment scenario X 9, Y 6, X 2, Y 1 and .6 What proportion should be invested in each fund in order to minimize the variance of the portfolio? 105