Daniel Grady is the financial advisor for a number of profes

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:

Daniel would like to minimize the beta of the stock portfolio (calculated using a weighted average of the amounts into the different stocks) while maintaining an expected return of at least 10.5%. Since future conditions may change, Daniel has decided that no more than 25% 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 proble. what are the expected return and beta for this portfolio

USE EXCEL

Solution

s a financial advisor our obejective is to minimize the beta on the stocks by adjusting the weights of investment in different stocks so weightage of investment in five different stocks would be our decision variables. Objective function will be to minimize the over all risk of the portfolio which is weight average of beta of all five stocks Constraints are wightage of any of the given stock is not more than 35% and expected weighted average return of port folio should be more than 11% and sum of weights of all five stock should be equal to 1. Formulation of Lenar Programming Model Decision Variable Stocks Expected return Beta Weights 1 11% 1.2 w1 2 9% 0.85 w2 3 6.50% 0.55 w3 4 15% 1.4 w4 5 13% 1.25 w5 Objective function : Minimize beta Portfolio beta = 1.2w1 + 0.85w2 + 0.55w3 + 1.4w4 + 1.25w5 0 Constraints Over all return on portfolio 0.11w1 +0.09w2 + 0.065w3 + 0.15w4 + 0.13w5 0 >= 11% w1 0 <= 35% w2 0 <= 35% w3 0 <= 35% w4 0 <= 35% w5 0 <= 35% w1+w2+w3+w4+w5 0 = 100% Optimal Solution by solving in excel solver Decision Variable Stocks Expected return Beta Weights 1 11% 1.2 w1 0.00% 2 9% 0.85 w2 10.63% 3 6.50% 0.55 w3 35.00% 4 15% 1.4 w4 35.00% 5 13% 1.25 w5 19.37% Objective function : Minimize beta Portfolio beta = 1.2w1 + 0.85w2 + 0.55w3 + 1.4w4 + 1.25w5 1.015 Constraints Over all return on portfolio 0.11w1 +0.09w2 + 0.065w3 + 0.15w4 + 0.13w5 11% >= 11% w1 0% <= 35% w2 11% <= 35% w3 35% <= 35% w4 35% <= 35% w5 19% <= 35% w1+w2+w3+w4+w5