Selecting an Investment Portfolio: An investment manager wants to determine an o

Question

Selecting an Investment Portfolio: An investment manager wants to determine an optimal portfolio for a wealthy client. The fund has $2.5 million to invest, and its objective is to maximize total dollar return from both growth and dividends over the course of the coming year. The client has researched eight high-tech companies and wants the portfolio to consist of shares in these firms only. Three of the firms (S1-S3) are primarily software companies, three (H1-H3) are primarily hardware companies, and two (C1-C2) are Internet consulting companies. The client has stipulated that no more than 40% of the investment be allocated to any one of these three sectors. To ensure diversification, at least $100,000 must be invested in each of the eight stocks. Moreover, the number of shares invested in any stock must be a multiple of 1000. The table below gives estimates from the investment company's database relating to these stocks. These estimates include the price per share, the projected annual growth rate in the share price, and the anticipated annual dividend payment per share. Determine the maximum return on the portfolio. What is the optimal number of shares to buy for each of the stocks? What is the corresponding dollar amount invested in each stock? Compare the solution in which there is no integer restriction on the number of shares invested. By how much (in percentage terms) do the integer restrictions alter the value of the optimal objective function? By how much (in percentage terms) do they alter the optimal investment quantities?

Explanation / Answer

Expected return of stock= D1 /P0 + g

S1 = 2(1+0.05) / 40 + 0.05 = 10.25

S2=1.50 (1+010) /50 + 0.10 = 13.3

S3= 3.50 (1+0.03)/ 80 + 0.03 = 7.51

H1=3. (1+0.04)/ 60 + 0.04 = 9.2

H2 =2 (1+0.07) /45 + 0.07 = 11.76

H3= 1 (1+0.15) /60 + 0.15 = 16.92

C1 = 1.8 (1+0.22)/ 30 + 0.22 = 29.32

C2 = 0 (1+0.25) /25 = 0

Maximum amount invested in 1 sector = 2.5 million * 40% = 1 million

Minimum investment in each stock = .1 million

Return on the protfolio if invested equally = 10.25 + 13.3 + 7.51 + 9.2 + 11.76 + 16.92 +29.32 + 0 / 8 = 12.28

40% investment in sector 3 (c1, c2) = C1 = 900000 , C2 = 100000

40% investment in sector 2 (H1 , H2, H3) = H1 =100000, H2 = 100000, H3 = 800000

Balance in sector 1 (S1, S2, S3 ) = S1 = 100000, S2= 300000 , S3 = 100000

Maximum return in portfolio = = 10.25*(.1/2.5) + 13.3 *(.3/2.5) + 7.51 *(.1/2.5) + 9.2* (.1/2.5) + 11.76 * (0.1/2.5)+ 16.92 * (.8/2.5) +29.32 * (.9/2.5)+ 0 * (.1/2.5) = 19.11 %

Optimal number of share to buy each of the stock =

S1 = 100000/ 40 = 2500

S2= 100000/ 50 = 2000

S3= 300000/ 80 = 3750

H1= 100000/ 60 = 1666.67

H2 = 100000/ 45 = 2222.22

H3= 800000/ 60 = 13333.33

C1 = 900000/ 30 = 30000

C2 = 100000/ 25 = 4000

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