An investor has up to $250,000 to invest in three types of investments. Type A pays 8% annually and has a risk factor of
0. Type B pays 10% annually and has a risk factor of 0.06.
Type C pays 14% annually and has a risk factor of 0.10. To
have a well-balanced portfolio, the investor imposes the following conditions. The average risk factor should be no
greater than 0.05. Moreover, at least one-fourth of the total
portfolio is to be allocated to Type A investments and at least
one-fourth of the portfolio is to be allocated to Type B investments. How much should be allocated to each type of investment to obtain a maximum return?
if the risk factors are as stated, I assume the average is weighted by the amount invested? If so, then you want to minimize
p = (0A + 0.06B + 0.10C)/3
If the weight also includes the interest rate, then include that as a factor in each term.
Now specify the other constraints, and solve as usual.
ansewer
To solve this problem and allocate the investments in a way that maximizes the return, we can use linear programming.
Let's define the following variables:
- Let x be the amount invested in Type A investments.
- Let y be the amount invested in Type B investments.
- Let z be the amount invested in Type C investments.
Now, let's set up the objective function and the constraints:
Objective Function: Maximize the total return.
R = 0.08x + 0.10y + 0.14z
Constraints:
1. The total investment amount is up to $250,000.
x + y + z ≤ 250,000
2. The average risk factor should be no greater than 0.05.
(0.00x + 0.06y + 0.10z) / (x + y + z) ≤ 0.05
3. At least one-fourth of the total portfolio should be allocated to Type A investments.
x ≥ (1/4) * (x + y + z)
4. At least one-fourth of the total portfolio should be allocated to Type B investments.
y ≥ (1/4) * (x + y + z)
5. Non-negativity constraint.
x, y, z ≥ 0
Once we set up these equations, we can use linear programming techniques such as simplex method or gradient descent to solve the problem and find the optimal values for x, y, and z that maximize the return.
Note: To solve it computationally, you can use software or programming languages that provide linear programming solvers such as Python's scipy library, Excel Solver, or MATLAB.