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 the income that they have 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 beta for five stocks are as follows:

STOCK $12345$

Expected return $(\%)1196\mathrm{.}51513$

Beta $1\mathrm{.}20\mathrm{.}850\mathrm{.}551\mathrm{.}41\mathrm{.}25$

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. The attached and files shows the solution for the problem in XLI Math Prog.

a$.$ Formulate this as a linear program. $($Hint: Define each variable as the proportion of the

total investment that would be put in that stock. Include a constraint that restricts the

sum of these variables to be $1.)$

b$.$ Use the above model and solve the problem. How much should be invested in each stock?

What is the risk?

c$.$ Daniel wants to add $3$ more stock options to the problem $($in addition to the $5$ existing

options$).$ The expected return and the Beta values are given below. Modify the above

model$/$data $($or your own LP Solve model$)$ and solve the problem again. How much should

be invested in each stock? What is the risk? $($Requirements do not change$)$

STOCK $678$

Expected return $(\%)121510$

Beta $1\mathrm{.}221\mathrm{.}51.$