=8-22 (Portfolio selection problem) Daniel Grady is the financial advisor for a number of professional athletes. An analysis of the long-term goals for many of these athletes has resulted in a recommendation to purchase stocks with some of their income that is set aside for investments. Five stocks have been identified as having very favorable expectations for future performance. Although the expected return is important in these investments, the risk, as measured by the beta of the stock, is also important. (A high value of beta indicates that the stock has a relatively high risk.) The expected return and the betas for five stocks are as follows:

Stock 1 2 3 4 5 Expected return (%) 11.0 9.0 6.5 15.0 13.0 Beta 1.20 0.85 0.55 1.40 1.25 Daniel would like to minimize the beta of the stock portfolio (calculated using a weighted average of the amounts put into the different stocks) while maintaining an expected return of at least 11%. Since future conditions may change, Daniel has decided that no more than 35% of the portfolio should be invested in any one stock.

(a) Formulate this as a linear program. (Hint: Define the variables to be the proportion of the total investment that would be put in each stock. Include a constraint that restricts the sum of these variables to be 1.)

(b) Solve this problem. What are the expected return and beta for this portfolio?