As an investment manager, you now want to determine an optimal portfolio for a w
ID: 2741185 • Letter: A
Question
As an investment manager, you now want to determine an optimal portfolio for a wealthy client from Asia. Your client has $2.5 million to invest, and his objective is to maximize total dollar return from both growth and dividends over the course of the coming year. Your 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. Your client has stipulated that no more than 40 percent of the investment be allocated to any one of these three sectors. To assure diversification, at least $100,000 must be invested in each of the eight stocks. Also, the number of shares invested in any stock must be a multiple of 100. The table below gives estimates from your 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. Stock S1 S2 S3 H1 H2 H3 C1 C2 Price per share $40 $50 $80 $60 $45 $60 $30 $25 Growth rate 0.05 0.1 0.03 0.04 0.07 0.15 0.22 0.25 Dividend $2.00 $1.50 $3.50 $3.00 $2.00 $1.00 $1.80 $0.00 a) Formulate an ILP problem to determine the maximum return on the portfolio. b) Implement your model in a spreadsheet and solve it. c) What is the optimal number of shares to buy for each of the stocks? What is the corresponding dollar amount invested in each stock? d) 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 term) do they alter the optimal investment quantities?
Explanation / Answer
A) ILP
Maximise return = 4*1+6.5*2+5.9*3+5.4*4+5.15*5+10*6+8.4*7+6.25*8
Particulars
S1
S2
S3
H1
H2
H3
C1
C2
Price/Share -( A )
40
50
80
60
45
60
30
25
Growth Rate(%) - ( B )
0.05
0.1
0.03
0.04
0.07
0.15
0.22
0.25
Dividend ( C )
2
1.5
3.5
3
2
1
1.8
0
Growth in $ (A)*(B) = (D)
2
5
2.4
2.4
3.15
9
6.6
6.25
Net Profit per year ( C ) + (D)
4
6.5
5.9
5.4
5.15
10
8.4
6.25
No of Shares
2500
6000
1300
1700
2300
13200
29900
4000
Investment in each company
100000
300000
104000
102000
103500
792000
897000
100000
Investment
2498500
Sum of each investment
Profit
485855
Industry S
504000
Industry H
997500
Industry C
997000
D) Change in Optimal Investment = 0.26%
S1
S2
S3
H1
H2
H3
C1
C2
No of Shares
2500
6000
1250
1666.667
2222.222
13333.33
30000
4000
Investment in each
100000
300000
100000
100000
100000
800000
900000
100000
C) Optimal number of shares and corresponding dollar investmetn
S1
S2
S3
H1
H2
H3
C1
C2
No of Shares
2500
6000
1300
1700
2300
13200
29900
4000
Investment in each
100000
300000
104000
102000
103500
792000
897000
100000
Particulars
S1
S2
S3
H1
H2
H3
C1
C2
Price/Share -( A )
40
50
80
60
45
60
30
25
Growth Rate(%) - ( B )
0.05
0.1
0.03
0.04
0.07
0.15
0.22
0.25
Dividend ( C )
2
1.5
3.5
3
2
1
1.8
0
Growth in $ (A)*(B) = (D)
2
5
2.4
2.4
3.15
9
6.6
6.25
Net Profit per year ( C ) + (D)
4
6.5
5.9
5.4
5.15
10
8.4
6.25
No of Shares
2500
6000
1300
1700
2300
13200
29900
4000
Investment in each company
100000
300000
104000
102000
103500
792000
897000
100000
Investment
2498500
Sum of each investment
Profit
485855
Industry S
504000
Industry H
997500
Industry C
997000
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