Portfolios and linear programming 6
Toy Model
There are four kinds of investments, each with an associated return and risk (1-4, 1 is the best).
| Investment | Return(%) | Risk |
|---|---|---|
| Stock A | 4 | 1 |
| Stock B | 5 | 2 |
| Bond A | 10 | 3 |
| Bond B | 20 | 4 |
Alice invests $1000 in stocks and bonds, the goal is to maximize the total return. The following requirements have to be satisfied.
a. The total risk should be no greater than 3000.
b. The total amount of money being invested should not exceed $1000.
c. The amount in stock A and in stock B minus bond A should be no greater than $500.
Decision Variables
X1= stock A
X2= stock B
X3= bond A
X4= bond B
Objective function
Maximize: Z=0.04X1+0.05X2+0.1X3+0.2X4
Constraints
X1+X2+X3+X4<= 1000
X1+2X2+3X3+4X4<=3000
X1+X2-X3<=500
X1, X2, X3<=0
Slack Variables
X5=1000-X1-X2-X3-X4
X6=3000-X1-2X2-3X3-4X4
X7=500-X1-X2+X3
Last dictionary
X4=750-0.25X1-0.5X2-0.75X3-0.25X6
X5=250-0.75X1-0.5X2-0.25X3+0.25X6
X7=500-X1-X2+X3
Z=150-0.01X1-0.05X2-0.05X3-0.05X6
Solution X1=0, X2=0, X3=0, X4=750, X5=250, X6=0, X7=500. The optimal value of Z is 150.
Dual problem
Objective function
Minimize: W=1000Y1+3000Y2+500Y3
Constraint
Y1+Y2+Y3>=0.04
Y1+2Y2+Y3>=0.05
Y1+3Y2-Y3>=0.1
Y1+4Y2>=0.2
Y1, Y2, Y3, Y4>=0
Solution
Y1=0, Y2=0.05, Y3=0. The optimal value of W is 150, it is the same as the toy model.
Results
The optimal value is 150, when X1=0, X2=0, X3=0, X4=750. This means when puting $750 in Bond B, it reaches the maximal profit, which is $ 150.
Y1: it refers to the first constraint. The first constraint is the upper bound of the total amount of investment. If we increase $1 to the investment, the maximal profit does not change.
Y2: it refers to the second constraint, which is the upper bound of the total risk. If we increase the total risk by 1, the maximal profit increase $0.05.
Y3: it refers to the third constraint. If we increase the third constraint by 1, the maximal profit does not change.
Actual Portfolio
| Variable | Name | Risk | YTD Return |
|---|---|---|---|
| x1 | HSBC MSCI Brazil ETF | 4 | -3.21 |
| x2 | HSBC MSCI Brazil ETF EUR | 4 | -4.64 |
| x3 | HSBC EURO STOXX 50 ETF | 4 | 13.45 |
| x4 | HSBC EURO STOXX 50 ETF EUR | 4 | 13.40 |
| x5 | HSBC EURO STOXX 50 ETF EUR | 4 | 11.64 |
| x6 | HSBC EURO STOXX 50 ETF GBP | 4 | 12.29 |
| x7 | HSBC EURO STOXX 50 ETF USD | 4 | 11.87 |
| x8 | HSBC FTSE 100 ETF | 3 | 12.87 |
| x9 | HSBC FTSE 100 ETF EUR | 3 | 11.77 |
| x10 | HSBC FTSE 100 ETF EUR | 3 | 12.40 |
| x11 | HSBC FTSE 100 ETF USD | 3 | 12.19 |
| x12 | HSBC FTSE 250 ETF | 2 | 21.08 |
| x13 | HSBC FTSE EPRA/NAREIT Developed ETF | 4 | 5.57 |
| x14 | HSBC FTSE EPRA/NAREIT Developed ETF EUR | 4 | 4.94 |
| x15 | HSBC FTSE EPRA/NAREIT Developed ETF GBP | 4 | 4.29 |
max -3.21x1-4.64x2+13.45x3+13.4x4+11.64x5+12.29x6+11.87x7+12.87x8+11.77x9+12.4x10+12.19x11+21.08x12+5.57x13+4.94x14+4.29x15
st x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15<=5000
4x1+4x2+4x3+4x4+4x5+4x6+4x7+3x8+3x9+3x10+3x11+2x12+4x13+4x14+4x15<=15000
Results
Using Lindo to solve the problem
LP OPTIMUM FOUND AT STEP 0
OBJECTIVE FUNCTION VALUE
1) 105400.0
VARIABLE VALUE REDUCED COST
X1 0.000000 24.290001
X2 0.000000 25.719999
X3 0.000000 7.630000
X4 0.000000 7.680000
X5 0.000000 9.440000
X6 0.000000 8.790000
X7 0.000000 9.210000
X8 0.000000 8.210000
X9 0.000000 9.309999
X10 0.000000 8.680000
X11 0.000000 8.890000
X12 5000.000000 0.000000
X13 0.000000 15.510000
X14 0.000000 16.139999
X15 0.000000 16.790001
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 21.080000
3) 5000.000000 0.000000
NO. ITERATIONS= 0
The maximal total profit is $104500.