An investment advisor must invest $400,000 for a client (all of this must be invested). The annual rate of return for the portfolio must be at least 10%, and no stock can account for more than 40% of the total investment. The four stocks being considered and the relevant financial data are as follows:

Stock/ Acme / Bravo/ Centaur /Delta

Price per share/ $100 / $50 / $80 / $40

Annual rate of return/ 0.12 /0.09/ 0.08 / 0.11

beta / 1.5/ 1.3 / 0.9 / 1.4

Beta is a measure of risk. A high beta indicates that the return on a stock is volatile. Thus, a low beta is usually desirable. The beta (or average beta) for a portfolio of stocks is a weighted average of the betas and the weights are determined by the amount of money that is put into each stock. For example, if $40,000 were put into each of the first two stocks, and $160,000 were put into each of the other two, the beta for the portfolio would be:

Total beta for portfolio =1.5(40,000) + 1.3(40,000) + 0.9(160,000) + 1.4(160,000) = 480,000

Average beta (or simply beta) = (total beta)/(total dollars invested) = 480,000/400,000 = 1.2

Linear programming is used to develop an investment portfolio that minimizes risk.

The variables are defined as:

A = $ invested in Acme

B = $ invested in Bravo

C = $ invested in Centaur

D = $ invested in Delta

Minimize total beta = 1.5A + 1.3B + 0.9C + 1.4D

Subject to:

A + B + C + D = 400,000 Total invested

0.12A + 0.09B + 0.08C + 0.11D > 40,000 Return is at least 10% of investment

A < 160,000 Maximum $ in Acme

B < 160,000 Maximum $ in Bravo

C < 160,000 Maximum $ in Centaur

D < 160,000 Maximum $ in Delta

A, B, C, D > 0

a. If the company invested $100,000 in each stock, calculate the number of shares, the beta (i.e. average beta) for the portfolio, and give the return as a percent. Put answers on answer sheet.

b. Find the optimal solution using computer software and fill in the table.

Note: The objective is expressed as the total beta and the return in constraint 2 is expressed in dollars. This is to avoid round-off errors. You will calculate the average beta (or simply beta) and the percent return after finding the optimal solution.

a.

Acme/ Bravo/ Centaur/ Delta

$ invested _________ / _________/ ________ / _________

Number of shares ________/ __________/ _________/ _________

Portfolio return (%)_________

Prtfolio beta ____________

b.

Acme/ Bravo/ Centaur/ Delta

$ invested _________ / _________/ ________ / _________

Number of shares ________/ __________/ _________/ _________

Portfolio return (%)_________

Prtfolio beta ____________