Question

In a portfolio problem, the investor has up to $50,000 to invest in stocks 1, 2, and 3, which have selling prices of $15/share, $47.25/share, and $110/share, respectively. The investor can purchase multiple shares of multiple stocks. The expected returns on investment of the three stocks are 6%, 8%, and 11%. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2, and the total number of shares of stocks 2 and 3 does not exceed 350. The investor stipulates that stock 1 must not account for more than 35% of the total number of shares purchased. How should the investor make the investment to maximize the return? Formulate this problem as a linear/integer programming model (not using Excel, no need to solve it).

Answer 1

Stock Price:

Stock 1: $15/share

Stock 2: $47.25 /share

Stock 3: $110 /share

Expected Return per share from Stock 1 = 6% of share price = 0.06*15 = $0.90

Expected Return per share from Stock 2 = 8% of share price = 0.08*47.25 = $3.78

Expected Return per share from Stock 3 = 11% of share price = 0.11*110 = $12.1

Decision Variable:

x1 = Number of Shares purchased of Stock 1

x2 = Number of Shares purchased of Stock 2

x3 = Number of Shares purchased of Stock 3

Objective Function:

Maximize Return (Z) = Sum of (No. of Shares * Expected Return)

Z = 0.90*x1 + 3.78*x2 + 12.1*x3

Constrains:

1.Total Investment <= 50000

Sum of (Stock Price*Number of Share) <= 50000

15*x1 + 47.25*x2 + 110*x3 <=50000

2. Investment in stock 2<=10000

47.25*x2 <= 10000

3. Total number of shares of stock 2 and 3 <= 350

x2 + x3 <=350

4. Number of Shares of Stock 1 <= 35% of Total number of shares

x1 <= 0.35*(x1 + x2 + x3)

0.65x1 - 0.35*x2 - 0.35*x3 <=0

5. x1,x2,x3 >= 0 (integers)