Math Problem Statement
Solution
This problem can be solved as a linear programming (optimization) problem.
Decision Variables:
- Let = amount invested in stocks (at 15% return).
- Let = amount invested in bonds (at 10% return).
- Let = amount invested in money market funds (at 8% return).
Objective Function:
Maximize total return:
Constraints:
-
Total investment cannot exceed $300,000:
-
At least $75,000 must be invested in money market funds:
-
The amount invested in bonds must be at least equal to the sum of the investments in stocks and money market funds:
-
Non-negativity constraint:
I'll solve this optimization problem to determine the optimal values for , , and that maximize .from scipy.optimize import linprog
Coefficients for the objective function (maximize returns)
c = [-0.15, -0.10, -0.08] # Negative because linprog performs minimization
Coefficients for the inequality constraints (Ax <= b)
A = [ [1, 1, 1], # Total investment <= 300,000 [-1, 1, -1], # x2 >= x1 + x3 --> -x1 + x2 - x3 >= 0 [0, 0, -1] # x3 >= 75,000 --> -x3 <= -75,000 ]
b = [300000, 0, -75000]
Bounds for each variable (x1, x2, x3 >= 0)
x_bounds = (0, None) # Stocks y_bounds = (0, None) # Bonds z_bounds = (75000, None) # Money Market Funds
Solve the linear program
result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds, z_bounds], method='highs')
Extract results
investment_amounts = result.x maximum_return = -result.fun
investment_amounts, maximum_return(array([225000., 0., 75000.]), 39750.0)(array([225000., 0., 75000.]), 39750.0)successThe optimal investment strategy is as follows:
- Stocks: $225,000
- Bonds: $0
- Money Market Funds: $75,000
The maximum return Sandra can achieve is $39,750.
Let me know if you would like a detailed explanation or further assistance.
Related Questions:
- How can the constraints be modified to include a specific investment ratio?
- What would happen if Sandra wanted a higher minimum for the money market funds?
- How can linear programming be applied to more complex financial planning?
- What if the expected returns for stocks and bonds were reversed?
- How can Sandra handle risk considerations with this model?
Tip:
Always verify whether an optimal solution meets all constraints to ensure feasibility in real-world scenarios.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Formulas
Objective Function: Maximize Z = 0.15x1 + 0.10x2 + 0.08x3
Constraints: x1 + x2 + x3 <= 300,000
x3 >= 75,000
x2 >= x1 + x3
x1, x2, x3 >= 0
Theorems
Linear Programming Feasibility Theorem
Suitable Grade Level
Undergraduate Level or Advanced High School
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