An investment advisor must invest $200,000 for a client (all of this must be inv
ID: 2935460 • Letter: A
Question
An investment advisor must invest $200,000 for a client (all of this must be invested). The annual rate of return for the portfolio must be at least 9.5%, 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
0.11
0.09
0.08
0.10
beta
1.7
1.4
1.0
1.5
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 $20,000 were put into each of the first two stocks, and $80,000 were put into each of the other two, the beta for the portfolio would be:
Total beta for portfolio =1.7(20,000) + 1.4(20,000) + 1.0(80,000) + 1.5(80,000) = 262,000
Average beta (or simply beta) = (total beta)/(total dollars invested) = 262,000/200,000 = 1.31
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 = 1.7A + 1.4B + 1.0C + 1.5D
Subject to:
A + B + C + D = 200,000 Total invested
0.11A + 0.09B + 0.08C + 0.10D > 19,000 Return is at least 9.5% of investment
A < 80,000 Maximum $ in Acme
B < 80,000 Maximum $ in Bravo
C < 80,000 Maximum $ in Centaur
D < 80,000 Maximum $ in Delta
A, B, C, D > 0
a. If the company invested $50,000 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.
Answer Sheet:
Acme
Bravo
Centaur
Delta
2 a.
$ invested
Number of shares
Portfolio return (%)
Portfolio beta
Acme
Bravo
Centaur
Delta
2 b.
$ invested
Number of shares
Portfolio return (%)
Portfolio beta
Stock
Acme
Bravo
Centaur
Delta
Price per share
$100
$50
$80
$40
Annual rate of return
0.11
0.09
0.08
0.10
beta
1.7
1.4
1.0
1.5
Explanation / Answer
Acme Bravo Centaur Delta 2A $ invested 50000 50000 50000 50000 Number of shares 500 1000 625 1250 Portfolio return (%) 9.50% Portfolio beta 1.4 Acme Bravo Centaur Delta 2 b. $ invested 60000 0 80000 60000 Number of shares 600 0 1000 1500 Portfolio return (%) 9.50% Portfolio beta 1.52
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